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LIBRARY 

OF  THE 

University  of  California. 


Class 


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THE 

HINDU-ARABIC  NUMERALS 


BY     . 

DAVID  EUGENE  SMITH 

AND 

LOUIS  CHARLES  KARPINSKI 


BOSTON  AND  LONDON 


GINN  AND  COMPANY,  PUBLISHERS 

1911 


OOPTinGHT.  1P11.  T.Y  PATTD    EUGENE    SMITH 

AND   LOUTS    CHAElEf  KAEPORSKI 

ATT,  EIGHT;  ED 

811.7 


a.A. 


PREFACE 

So  familiar  are  we  with  the  numerals  that  bear  the 
misleading  name  of  Arabic,  and  so  extensive  is  their  use 
in  Europe  and  the  Americas,  that  it  is  difficult  for  us  to 
realize  that  their  general  acceptance  in  the  transactions 
of  commerce  is  a  matter  of  only  the  last  four  centuries, 
and  that  they  are  unknown  to  a  very  large  part  of  the 
human  race  to-day.  It  seems  strange  that  such  a  labor- 
saving  device  should  have  struggled  for  nearly  a  thou- 
sand years  after  its  system  of  place  value  was  perfected 
before  it  replaced  such  crude  notations  as  the  one  that 
the  Roman  conqueror  made  substantially  universal  in 
Europe.  Such,  however,  is  the  case,  and  there  is  prob- 
ably no  one  who  has  not  at  least  some  slight  passing 
interest  in  the  story  of  this  struggle.  To  the  mathema 
tician  and  the  student  of  civilization  the  interest  is  gen- 
erally a  deep  one ;  to  the  teacher  of  the  elements  of 
knowledge  the  interest  may  be  less  marked,  but  never- 
theless it  is  real ;  and  even  the  business  man  who  makes 
daily  use  of  the  curious  symbols  by  which  we  express 
the  numbers  of  commerce,  cannot  fail  to  have  some 
appreciation  for  the  story  of  the  rise  and  progress  of 
these  tools  of  his  trade. 

This  story  has  often  been  told  in  part,  but  it  is  a  long 
tune  since  any  effort  has  been  made  to  bring  together 
the  fragmentary  narrations  and  to  set  forth  the  gen- 
eral problem  of  the  origin  and  development  of   these 

iii 

236299 


I 


iv  THE  HINDU-ARABIC  NUMERALS 

numerals.  In  this  little  work  we  have  attempted  to  state 
the  history  of  these  forms  in  small  compass,  to  place 
before  the  student  materials  for  the  investigation  of  the 
problems  involved,  and  to  express  as  clearly  as  possible 
the  results  of  the  labors  of  scholars  who  have  studied 
the  subject  in  different  parts  of  the  Avorld.  We  have 
had  no  theory  to  exploit,  for  the  history  of  mathematics 
has  seen  too  much  of  this  tendency  already,  but  as  far 
as  possible  we  have  weighed  the  testimony  and  have  set 
forth  what  seem  to  be  the  reasonable  conclusions  from 
the  evidence  at  hand. 

To  facilitate  the  work  of  students  an  index  has  been 
prepared  which  we  hope  may  be  serviceable.  In  this  the 
names  of  authors  appear  only  when  some  use  has  been 
made  of  then-  opinions  or  when  their  works  are  first 
mentioned  in  full  in  a  footnote. 

If  this  work  shall  show  more  clearly  the  value  of  our 
number  system,  and  shall  make  the  study  of  mathematics 
seem  more  real  to  the  teacher  and  student,  and  shall  offer 
material  for  interesting  some  pupil  more  fully  in  his  work 
with  numbers,  the  authors  will  feel  that  the  considerable 
labor  involved  in  its  preparation  has  not  been  in  vain. 

We  desire  to  acknowledge  our  especial  indebtedness 

to  Professor  Alexander  Ziwet  for  reading  all  the  proof, 

as  well  as  for  the  digest  of  a  Russian  work,  to  Professor 

Clarence  L.  Meader  for  Sanskrit  transliterations,  and  to 

Mr.  Steven  T.  Byington  for  Arabic  transliterations  and 

the  scheme  of  pronunciation  of  Oriental  names,  and  also 

our  indebtedness  to  other  scholars  in  Oriental  learning 

for  information. 

DAVID  EUGENE  SMITH 

LOUIS  CHARLES  KAR1TNSKI 


CONTENTS 


CHAPTER  PAGE 

PRONUNCIATION  OF  ORIENTAL  NAMES vi 

I.   EARLY  IDEAS  OF  THEIR  ORIGIN        1 

II.   EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE       .     .  12 

III.  LATER  HINDU  FORMS,  WITH  A  PLACE  VALUE    ...  38 

IV.  THE  SYMBOL  ZERO 51 

V.    THE    QUESTION     OF    THE    INTRODUCTION    OF    THE 

NUMERALS  INTO  EUROPE  BY  BOETHIUS    ....  63 
VI.    THE  DEVELOPMENT  OF  THE  NUMERALS  AMONG  THE 

ARABS 91 

VII.    THE  DEFINITE  INTRODUCTION   OF   THE  NUMERALS 

INTO  EUROPE 99 

VIII.    THE  SPREAD  OF  THE  NUMERALS  IN  EUROPE     ...  128 

INDEX 153 


PRONUNCIATION  OF  ORIENTAL  NAMES 

(S)  =  in  Sanskrit  names  and  words ;   (A)  =  in  Arabic  names  and  words. 


b,  d,  f,  g,  h,  j,  1,  m,  n,  p,  sh  (A),  t, 
th  (A),  v,  w,  x,  z,  as  in  English. 

a,  (S)  like  u  in  but:  thus  pandit, 
pronounced  pundit.  (A)  like  a  in 
ask  or  in  man.  a,  as  in  father. 

c,  (S)  like  ch  in  church  (Italian  c  in 
cento). 

d,  n,  s,  t,  (S)  d,  n,  ah,  t,  made  with 
the  tip  of  the  tongue  turned  up 
and  back  into  the  dome  of  the 
palate,  d,  s,  t,  z,  (A)  d,  s,  t,  z, 
made  with  the  tongue  spread  so 
that  the  sounds  are  produced 
largely  against  the  side  teeth. 
Europeans  commonly  pronounce 
d,  n,  s,  t,  z,  both  (S)  and  (A),  as 
simple  d,  n,  sh  (S)  or  s  (A),  t,  z. 
d  (A),  like  th  in  this. 

e,  (S)  as  in  they.    (A)  as  in  bed. 

g,  (A)  a  voiced  consonant  formed 
below  the  vocal  cords;  its  sound 
is  compared  by  some  to  a  g,  by 
others  to  a  guttural  r;  in  Arabic 
words  adopted  into  English  it  is 
represented  by  gh  (e.g.  ghoul), 
less  often  r  (e.g.  razzia). 

h  preceded  by  b,  c,  t,  t,  etc.  docs 
not  form  a  single  sound  with  these 
letters,  but  is  a  more  or  less  dis- 
tinct h  sound  following  them  ;  cf. 
the  sounds  in  abhor,  boathook, 
etc.,  or,  more  accurately  for  (S), 
the  "  bhoys  "  etc.  of  Irish  brogue. 
h  (A)  retains  its  consonant  sound 
at  the  end  of  a  word,  h,  (A)  an 
unvoiced  consonant  formed  below 
the  vocal  cords  ;  its  sound  is  some- 
times compared  to  German  hard 
ch,  and  may  be  represented  by  an 
h  as  strong  as  possible.  In  Arabic 
words  adopted  into  English  it  is 
represented   by  h,  e.g.   in  sahib, 


hakeem,  b.  (S)  is  final  consonant 
h,  like  final  h  (A). 

i,  as  in  pin.  I,  as  in  pique. 

k,  as  in  kick. 

kh,  (A)  the  hard  ch  of  Scotch  loch, 
German  ach,  especially  of  German 
as  pronounced  by  the  Swiss. 

m,  n,  (S)  like  French  final  m  or  n, 
nasalizing  the  preceding  vowel. 

n,  see  d.   n,  like  ng  in  .singing. 

o,  (S)  as  in  so.    (A)  as  in  obey. 

q,  (A)  like  k  (or  c)  in  cook;  further 
back  in  the  mouth  than  in  kick. 

r,  (S)  English  r,  smooth  and  on- 
trilled.  (A)  stronger.  r,(S)rused 
as  vowel,  as  in  apron  when  pro 
nounced  aprn  and  not  opera;  mod- 
ern Hindus  say  ri,  hence  our  am- 
rita,  Krishna,  for  a-mrta,  Krsna. 

s,  as  in  same,  s,  see  d.  §,  (S)  Eng- 
lish .s/i  (German  sch) . 

t,  see  d. 

u,  as  in  put.   u,  as  in  rule. 

y,  as  in  you. 

z,  see  d. 

',  (A)  a  sound  kindred  to  the  spiritus 
lenis  (that  is,  to  our  ears,  the  mere 
distinct  separation  of  a  vowel  from 
the  preceding  sound,  as  at  the  be- 
ginning of  a  word  in  German)  and 
to  h.  The '  is  a  very  distinct  sound 
in  Arabic,  but  is  more  nearly 
represented  by  the  spiritus  lenis 
than  by  any  sound  that  we  can 
produce  without  much  special 
training.  That  is,  it  should  be 
treated  as  silent,  but  the  sounds 
that  precede  and  follow  it  should 
not  run  together.  In  Arabic  words 
adopted  into  English  it  is  treated 
as  silent,  e.g.  in  Arab,  amber, 
Caaba  ('Arab,  'anbar,  ka'dbah). 


(A)  A  final  long  vowel  is  shortened  before  al  (7)  or  ibn  (whose  i  is  then 

silent). 

Accent  :  (S)  as  if  Latin  ;  in  determining  the  place  of  the  accent  m  and 
//  count  as  consonants,  but  //  after  another  consonant  does  not.  (A),  on 
the  last  syllable  that  contains  a  long  vowel  or  a  vowel  followed  by  two 
consonants,  except,  that  a  final  long  vowel  is  not  ordinarily  accented;  if 
there  is  no  long  vowel  nor  two  consecutive  consonants,  the  accent  falls  on 
the  first  syllable.    The  words  al  and  ibn  are  never  accented. 

vi 


J    >       > 


1  -J      )         3    > 


THE 

HINDU- ARABIC  NUMERALS 

CHAPTER  I 

EARLY  IDEAS   OF   THEIR  ORIGIN 

It  has  long  been  recognized  that  the  common  numerals 
used  in  daily  life  are  of  comparatively  recent  origin. 
The  number  of  systems  of  notation  employed  before 
the  Christian  era  wag  about  the  same  as  the  number  of 
written  languages,  and  in  some  cases  a  single  language 
had  several  systems.  The  Egyptians,  for  example,  had 
three  systems  of  writing,  with  a  numerical  notation  for 
each;  the  Greeks  had  two  well-defined  sets  of  numerals, 
and  the  Roman  symbols  for  number  changed  more  or  less 
from  century  to  century.  Even  to-day  the  number  of 
methods  of  expressing  numerical  concepts  is  much 
greater  than  one  would  believe  before  making  a  study 
of  the  subject,  for  the  idea  that  our  common  numerals 
are  universal  is  far  from  being  correct.  It  will  be  well, 
then,  to  think  of  the  numerals  that  we  still  commonly 
call  Arabic,  as  only  one  of  many  systems  in  use  just 
before  the  Christian  era.  As  it  then  existed  the  system 
was  no  better  than  many  others,  it  was  of  late  origin,  it 
contained  no  zero,  it  was  cumbersome  and  little  used, 

l 


;'■%  :  :•  :  ,'■':    Till:  1UNPU- ARABIC  NUMERALS 

and  it  bad  no  particular  promise.  Not  until  centuries  later 
did  the  system  have  any  standing  in  the  world  of  busi- 
ness and  science ;  and  had  the  place  value  which  now 
characterizes  it,  and  which  requires  a  zero,  been  worked 
out  in  Greece,  we  might  have  been  using  Greek  numerals 
to-day  instead  of  the  ones  with  which  we  are  familiar. 

Of  the  first  number  forms  that  the  world  used  this  is 
not  the  place  to  speak.  Many  of  them  are  interesting, 
but  none  had  much  scientific  value.  In  Europe  the  in- 
vention of  notation  was  generally  assigned  to  the  eastern 
shores  of  the  Mediterranean  until  the  critical  period  of 
about  a  century  ago,  —  sometimes  to  the  Hebrews,  some- 
times to  the  Egyptians,  but  more  often  to  the  early 
trading  Phoenicians.1 

The  idea  that  our  common  numerals  are  Arabic  in 
origin  is  not  an  old  one.  The  mediaeval  and  Renaissance 
writers  generally  recognized  them  as  Indian,  and  many 
of  them  expressly  stated  that  they  were  of  Hindu  origin.2 

1  "  Discipulus.  Quis  primus  invenit  numerurn  apud  Hebrreos  et 
iEgyptios  ?  Magister.  Abraham  primus  invenit  numerurn  apud 
Hebneos,  deinde  Moses  ;  et  Abraham  tradidit  istam  scientiam  numeri 
ail  JEgyptios,  et  docuit  eos :  deinde  Josephus."  [Bede,  De  computo 
dialogus  (doubtfully  assigned  to  him),  Opera  omnia,  Paris,  1802,  Vol.  I, 
1>.  650.] 

"  Alii  ref  erunt  ad  Phoenices  inventores  arithmetics,  propter  eandem 
commerciorum  caussam  :  Alii  ad  Indos  :  Ioannes  de  Sacrobosco,  cujus 
sepulehrum  est  Lutetiae  in  comitio  Maturinensi,  refert  ad  Arabes." 
[Ramus,  Arithmetical  libri  dvo,  Basel,  1509,  p.  112.] 

Similar  notes  are  given  by  Peletarius  in  his  commentary  on  the 
arithmetic  "!'  Gemma  Frisius  (1503  ed.,  fol.  77),  and  in  his  own  work 
(1570  Lyons  ed.,  p.  14)  :  "La  valeur  des  Figures  commence  au  coste 
dextre  tiranl  vers  le  coste  senestre :  au  rebours  de  notre  maniere 
d'escrire  par  re  que  la  premiere  prattique  est  venue  des  Chaldees : 
on  des  Pheniciens,  qui  out  6t6  les  premiers  traffiquers  de  marchan- 
dise." 

2  Maxiinus  Planudes  (C.  1330)  states  that  "the  nine  symbols  come 
from  the  Indians."    [Waschke's   German   translation,   Halle,   1878, 


EARLY  IDEAS  OE  THEIR  ORIGIN  3 

Others  argued  that  they  were  probably  invented  by  the 
Chaldeans  or  the  Jews  because  they  increased  in  value 
from  right  to  left,  an  argument  that  would  apply  quite 
as  well  to  the  Roman  and  Greek  systems,  or  to  any 
other.  It  was,  indeed,  to  the  general  idea  of  notation 
that  many  of  these  writers  referred,  as  is  evident  from 
the  words  of  England's  earliest  arithmetical  textbook- 
maker,  Robert  Recorde  (c.  1542):  "In  that  thinge  all 
men  do  agree,  that  the  Chaldays,  whiche  fyrste  inuented 
thys  arte,  did  set  these  figures  as  thei  set  all  their  letters. 
for  they  wryte  backwarde  as  you  tearme  it,  and  so  doo 
they  reade.  And  that  may  appeare  in  all  Hebrewe, 
Chaldaye  and  Arabike  bookes  .  .  .  where  as  the  Greekes, 
Latines,  and  all  nations  of  Europe,  do  wryte  and  reade 
from  the  lefte  hand  towarde  the  ryghte."  1    Others,  and 

p.  3.]  Willichius  speaks  of  the  "  Zyphrre  Indicpe,"  in  his  Arithmetical 
libri  tres  (Strasburg,  1540,  p.  93),  and  Cataneo  of  "  le  noue  figure  de 
gli  Indi,"  in  his  Le  pratiche  delle  dve  prime  mathematiche  (Venice,  1540, 
fol.  1).  Woepcke  is  not  correct,  therefore,  in  saying  ("M^moire  sur  la 
propagation  des  chiffres  indiens,"  hereafter  referred  to  as  Propagation 
[Journal  Asiatique,  Vol.  I  (0),  1803,  p.  34])  that  Wallis  (A  Treatise  on 
Algebra,  both  historical  and  practical,  London,  1085,  p.  13,  and  Be 
algebra  tractatus,  Latin  edition  in  his  Opera  omnia,  1093,  Vol.  II, 
p.  10)  was  one  of  the  first  to  give  the  Hindu  origin. 

1  From  the  1558  edition  of  The  Grovnd  o/Artes,  fol.  C,  5.  Similarly 
Bishop  Tonstall  writes  :  "  Qui  a  Chaldeis  priinum  in  finitimos,  deinde 
in  omnes  pene  gentes  fiuxit.  .  .  .  Numerandi  artem  a  Chaldeis  esse 
profectam  :  qui  dum  scribunt,  a  dextra  incipiunt,  et  in  leuam  pro- 
grediuntur."  [De  arte  supputandi,  London,  1522,  fol.  B,  3.]  Gemma 
Frisius,  the  great  continental  rival  of  Recorde,  had  the  same  idea : 
"  Prinium  autem  appellamus  dexterum  locum,  eo  quod  haec  ars  vel  a 
Chaldaeis,  vel  ab  Hebrajis  ortum  habere  credatur,  qui  etiam  eo  ordine 
scribunt"  ;  but  this  refers  more  evidently  to  the  Arabic  numerals. 
[Arithmetical  practical  methodvs  facilis,  Antwerp,  1540,  fol.  4  of  the 
1563  ed.]  Sacrobosco  (c.  1225)  mentions  the  same  thing.  Even  the 
modern  Jewish  writers  claim  that  one  of  their  scholars,  Mashallah 
(c.  800),  introduced  them  to  the  Mohammedan  world.  [C.  Levias, 
The  Jewish  Encyclopedia,  New  York,  1905,  Vol.  IX,  p.  348.] 


4  THE  HINDU-ARABIC  NUMERALS 

among  them  such  influential  writers  as  Tartaglia 1  in 
Italy  and  Kobel 2  in  Germany,  asserted  the  Arabic  origin 
of  the  numerals,  while  still  others  left  the  matter  unde- 
cided 3  or  simply  dismissed  them  as  "  barbaric." 4  Of 
course  the  Arabs  themselves  never  laid  claim  to  the  in- 
vention, always  recognizing  their  indebtedness  to  the 
Hindus  both  for  the  numeral  forms  and  for  the  distin- 
guishing feature  of  place  value.  Foremost  among  these 
writers  was  the  great  master  of  the  golden  age  of  Bag- 
dad, one  of  the  first  of  the  Arab  writers  to  collect  the 
mathematical  classics  of  both  the  East  and  the  West,  pre- 
serving them  and  finally  passing  them  on  to  awakening 
Europe.  This  man  was  Mohammed  the  Son  of  Moses, 
from  Khowarezm,  or,  more  after  the  manner  of  the  Arab, 
Mohammed  ibn  Musa  al-Khowarazml,5  a  man  of  great 

1  "...  &  que  esto  fu  trouato  di  fare  da  gli  Arabi  con  diece  figure.11 
[La  prima  parte  del  general  trattato  di  nvmeri,  et  misvre,  Venice,  1556, 
fol.  9  of  the  1592  edition.] 

2  "  Vom  welchen  Arabischen  audi  disz  Kunst  entsprungen  ist." 
[Ain  nerv  geordnet  llechenbiechlin,  Augsburg,  1514,  fol.  13  of  the  1531 
edition.  The  printer  used  the  letters  rv  for  w  in  "new11  in  the  first 
edition,  as  he  had  no  10  of  the  proper  font.] 

8  Among  them  Glareanus  :  "  Characteres  simplices  sunt  nouem  sig- 
nificatiui,  ab  Indis  usque,  siue  Chaldseis  asciti  .1.2.3.4.5.6.7.8.9.  Est 
item  unns  .0  circulus,  qui  nihil  significat.11  [De  VI.  Arithmeticae 
practicae  speciebvs,  Paris,  1539,  fol.  9  of  the  1543  edition.] 

4  "  Barbarische  oder  gemeine  Ziffern."  [Anonymous,  Das  Einmahl 
Eins  cum  notis  variorum,  Dresden,  1 703,  p.  3.]  So  Vossius  (I)e  universac 
matheseos  natura  et  constitutione  liber,  Amsterdam,  1650,  p.  34)  calls 
them  "Barbaras  numeri  notas."  The  word  at  that  time  was  possibly 
synonymous  with  Arabic. 

6  His  full  name  was  'Abu  'Abdallah  Mohammed  ibn  Musa  al- 
Khow&razml.  He  was  born  in  Khowarezm,  "the  lowlands,11  the 
country  about  the  present  Khiva  and  bordering  on  the  Oxus,  and 
lived  at  Bagdad  under  the  caliph  al-Mamun.  He  died  probably  be- 
tween 220  and  230  of  the  Mohammedan  era,  that  is,  between  835  and 
845  a. i).,  although  some  put  the  date  as  early  as  812.  The  best  ac- 
count of  this  great  scholar  may  be  found  in  an  article  by  C.  Nallino. 
"Al-HuwSrizipi,"  in  the  J. Mi  delta  K.Accad.  dei  Lined,  Rome,  1896.  See 


EARLY  IDEAS  OF  THEIR  ORIGIN  5 

learning1  and  one  to  whom  the  world  is  much  indebted 
for  its  present  knowledge  of  algebra 1  and  of  arithmetic. 
Of  him  there  will  often  be  occasion  to  speak  ;  and  in  the 
arithmetic  which  he  wrote,  and  of  which  Adelhard  of 
Bath2  (c.  1130)  may  have  made  the  translation  or  para- 
phrase,3 he  stated  distinctly  that  the  numerals  were  due 
to  the  Hindus.4  This  is  as  plainly  asserted  by  later  Arab 

also  Verhandlungcn  des  5.  Congresses  der  Orientalisten,  Berlin,  1882, 
Vol.  II,  p.  19;  W.  Spitta-Bey  in  the  Zeitschrift  der  deutschen  Morgen- 
liind.  Gesellschaft,  Vol.  XXXIII,  p.  224 ;  Steinschneider  in  the  Zeit- 
scltrift  der  deutschen  Morgenland.  Gesellschaft,  Vol.  L,  p.  214;  Treutlein 
in  the  Abhandlungen  zur  Geschichte  der  Mathematik,  Vol.  I,  p.  5 ;  Suter, 
"Die  Mathematiker  und  Astronomen  der  Araber  und  ihre  Werke," 
Abhandlungen  zur  Geschichte  der  Mathematik,Vol.  X,  Leipzig,  1900,  p.  10, 
and  "Nachtrage,"  in  Vol.  XIV,  p.  158 ;  Cantor,  Geschichte  der  Mathe- 
matik,Vol.  I,  3d  ed.,  pp.  712-733  etc. ;  F. Woepcke  in  Propagation,  p.  489. 
So  recently  has  he  become  known  that  Heilbronner,  writing  in  1742, 
merely  mentions  him  as  "  Ben-Musa,  inter  Arabes  Celebris  Geometra, 
scripsit  de  flguris  planis  &  sphericis."  [Historia  matheseos  universal, 
Leipzig,  1742,  p.  438.] 

In  this  work  most  of  the  Arabic  names  will  be  transliterated  sub- 
stantially as  laid  down  by  Suter  in  his  work  Die  Mathematiker  etc., 
except  where  this  violates  English  pronunciation.  The  scheme  of  pro- 
nunciation of  oriental  names  is  set  forth  in  the  preface. 

1  Our  word  algebra  is  from  the  title  of  one  of  his  works,  Al-jabr  waH- 
muqabalah,  Completion  and  Comparison.  The  work  was  translated  into 
English  by  F.  Rosen,  London,  1831,  and  treated  in  L'Algebre  d'al- 
Kharizmi  et  les  methodes  indienne  et  grecque,  Le"on  Rodet,  Paris,  1878, 
extract  from  the  Journal  Asialique.  For  the  derivation  of  the  word 
algebra,  see  Cossali,  Scritti  Inediti,  pp.  381-383,  Rome,  1857;  Leo- 
nardo's Liber  Abbaci  (1202),  p.  410,  Rome,  1857  ;  both  published  by  B. 
Boncompagni.    "Almuchabala"  also  was  used  as  a  name  for  algebra. 

2  This  learned  scholar,  teacher  of  O'Creat  who  wrote  the  Ilelceph 
("  Prologus  N.  Ocreati  in  Ilelceph  ad  Adelardum  Batensem  magistrum 
suwrn'1),  studied  in  Toledo,  learned  Arabic,  traveled  as  far  east  as 
Fgypt,  and  brought  from  the  Levant  numerous  manuscripts  for  study 
and  translation.  See  Henry  in  the  Abhandlungen  zur  Geschichte  der 
Mathematik,  Vol.  Ill,  p.  131 ;  Woepcke  in  Propagation,  p.  518. 

3  The  title  is  Algoritmi  de  numero  Indorum.  That  he  did  not  make 
this  translation  is  asserted  by  Enestrom  in  the  Bibliotheca  Mathematica, 
Vol.  I  (3),  p.  520. 

4  Thus  he  speaks  "de  numero  indorum  per  .IX.  literas,"  and  pro- 
ceeds :  "  Dixit  algoritmi :  Cum  uidissem  yndos  constituisse  .IX.  literas 


6  THE  HINDU-ARABIC  NUMERALS 

writers,  even  to  the  present  clay.1  Indeed  the  phrase 
'ibn  hindi,  "Indian  science,"  is  used  by  thern  for  arith- 
metic, as  also  the  adjective  hindl  alone.2 

Probably  the  most  striking  testimony  from  Arabic 
sources  is  that  given  by  the  Arabic  traveler  and  scholar 
Mohammed  ibn  Ahmed,  Abu  '1-Rihan  al-B  Irani  (973- 
1048),  who  spent  many  years  in  Hindustan.  He  wrote 
a  large  work  on  India,3  one  on  ancient  chronology,4  the 
"  Book  of  the  Ciphers,"  unfortunately  lost,  which  treated 
doubtless  of  the  Hindu  art  of  calculating,  and  was  the 
author  of  numerous  other  works.  Al-Blruni  was  a  man 
of  unusual  attainments,  being  versed  in  Arabic,  Persian, 
Sanskrit,  Hebrew,  and  Syriac,  as  well  as  in  astronomy, 
chronology,  and  mathematics.  In  his  work  on  India  he 
gives  detailed  information  concerning  the  language  and 

in  uniuerso  numero  suo,  propter  dispositionem  suam  quain  posuerunt, 
uolui  patefacere  de  opera  quod  fit  per  eas  aliquid  quod  esset  leuius 
discentibus,  si  deus  uoluerit."  [Boncompagni,  Trattati  d' Aritmetica, 
Rome,  1857.]  Discussed  by  F.  Woepcke,  Sur  V introduction  de  Varith- 
mttique  indienne  en  Occident,  Rome,  1859. 

1  Thus  in  a  commentary  by  'AH  ibn  Abi  Bekr  ibn  al-Jamal  al-Ansarl 
al-Mekki  on  a  treatise  on  gobar  arithmetic  (explained  later)  called  Al- 
murshidah,  found  by  Woepcke  in  Paris  (Propagation,  p.  G6),  there  is 
mentioned  the  fact  that  there  are  "nine  Indian  figures"  and  "a  sec- 
ond kind  of  Indian  figures  .  .  .  although  these  are  the  figures  of  the 
gobar  writing."  So  in  a  commentary  by  Hosein  ibn  Mohammed  al- 
Mahalll  (died  in  1756)  on  the  Mokhtasar  fl'ilm  el-hisah  (Extract  from 
Arithmetic)  by  "Abdalqadir  ibn'Ali  al-Sakhawi  (died  c.  1000)  it  is  re- 
lated that  "  the  preface  treats  of  the  forms  of  the  figures  of  Hindu 
signs,  such  as  were  established  by  the  Hindu  nation."  [Woepcke, 
Propagation,  p.  63.] 

2  See  also  Woepcke,  Propagation,  p.  505.  The  origin  is  discussed  at 
much  length  by  G.  11.  Kaye,  "Notes  on  Indian  Mathematics.  —  Arith- 
metical Notation,"  Journ.  and  Proc.  of  the  Asiatic  Soc.  of  Bengal,  Vol. 
Ill,  1907,  p.  489. 

3  Alberuni's  India,  Arabic  version,  London,  1887;  English  transla- 
tion, ibid.,  1888. 

4  Chronology  of  Ancient  Nations,  London,  1879.  Arabic  and  English 
versions,  by  C.  E.  Sachau. 


EARLY  IDEAS  OF  THEIR  ORIGIN  7 

customs  of  the  people  of  that  country,  and  states  ex- 
plicitly l  that  the  Hindus  of  his  time  did  not  use  the 
letters  of  their  alphabet  for  numerical  notation,  as  the 
Arabs  did.  He  also  states  that  the  numeral  signs  called 
ahka2  had  different  shapes  in  various  parts  of  India,  as 
was  the  case  with  the  letters.  In  his  Chronology  of  An- 
cient Nations  he  gives  the  sum  of  a  geometric  progression 
and  shows  how,  in  order  to  avoid  any  possibility  of  error, 
the  number  may  be  expressed  in  three  different  systems :  • 
with  Indian  symbols,  in  sexagesimal  notation,  and  by  an 
alphabet  system  which  will  be  touched  upon  later.  He 
also  speaks3  of  "179,  876,  755,  expressed  in  Indian 
ciphers,"  thus  again  attributing  these  forms  to  Hindu 
sources. 

Preceding  Al-BlrunJ  there  was  another  Arabic'  writer 
of  the  tenth  century,  Motahhar  ibn  Tahir,4  author  of 
the  Book  of  the  Creation  and  o'f  History,  who  gave  as  a 
curiosity,  in  Indian  (Nagari)  symbols,  a  large  number 
asserted  by  the  people  of  India  to  represent  the  duration 
of  the  world.  Huart  feels  positive  that  in  Motahhar's 
time  the  present  Arabic  symbols  had  not  yet  come  into 
use,  and  that  the  Indian  symbols,  although  known  to 
scholars,  were  not  current.  Unless  this  were  the  case, 
neither  the  author  nor  his  readers  would  have  found 
anything  extraordinary  in  the  appearance  of  the  number 
which  he  cites. 

Mention  should  also  be  made  of  a  widely-traveled 
student,  Al-Mas'udl  (885  ?~956),  whose  journeys  carried 
him  from  Bagdad  to  Persia,  India,  Ceylon,  and  even 

1  India,  Vol.  I,  chap.  xvi. 

2  The  Hindu  name  for  the  symbols  of  the  decimal  place  system. 

3  Sachau's  English  edition  of  the  Chronology,  p.  64. 

4  Literature  arabe,  CI.  Huart,  Paris,  1902. 


8  THE  HINDU-ARABIC  NUMERALS 

across  the  China  sea,  and  at  other  thnes  to  Madagascar, 
Syria,  and  Palestine.1  He  seems  to  have  neglected  no 
accessible  sources  of  information,  examining  also  the 
history  of  the  Persians,  the  Hindus,  and  the  Romans. 
Touching  the  period  of  the  Caliphs  his  work  entitled 
Meadows  of  Gold  furnishes  a  most  entertaining  fund  of 
information.  He  states2  that  the  wise  men  of  India, 
assembled  by  the  king,  composed  the  Sindhind.  Fur- 
ther on  3  he  states,  upon  the  authority  of  the  historian 
Mohammed  ibn  'AH  'Abdi,  that  by  order  of  Al-Mansur 
many  works  of  science  and  astrology  were  translated  into 
Arabic,  notably  the  Sindhind  (Siddhdnta).  Concerning 
the  meaning  and  spelling  of  this  name  there  is  consider- 
able diversity  of  opinion.  Colebrooke  4  first  pointed  out 
the  connection  between  Siddhdnta  and  Sindhind.  He 
ascribes  to  the  word  the  meaning  "  the  revolving  ages."  5 
Similar  designations  are  collected  by  Sedillot,6  who  in- 
clined to  the  Greek  origin  of  the  sciences  commonly 
attributed  to  the  Hindus.7  Casiri,8  citing  the  Tdr'ikh  al- 
hokamd  or  Chronicles  of  the  Learned,0  refers  to  the  work 

1  Huart,  History  of  Arabic  Literature,  English  ed.,  New  York,  1903, 
p.  182  seq. 

2  Al-Mas'udi's  Meadows  of  Gold,  translated  in  part  by  Aloys  Spren- 
ger,  London,  1841  ;  Les prairies  d'or,  trad,  par  C.  Barbier  de  Meynard 
et  Pavet  de  Courteille,  Vols.  I  to  IX,  Paris,  1801-1877. 

3  Les  prairies  d'or,  Vol.  VIII,  p.  289  seq. 
*  Essays,  Vol.  II,  p.  428. 

5  Loc.  cit.,  p.  504. 

8  Mat&riaux  pour  scrvir  a  Vhistoire  compar6e  de$  sciences  maiMma- 
tiques  chez  les  Grecs  et  les  Orientaux,  2  vols.,  Paris,  1845-1849,  pp.  438- 
439. 

7  lb'  made  an  excepl  ion,  however,  in  favor  of  the  numerals,  Inc.  cit.. 
Vol.  II.  p.  :,()3. 

8  Bibliotheca  Arabico-Hispana  Escurialensis,  Madrid,  1 7(>(>-l  770, 
pp.   126   127. 

9  The  author,  Ibn  al-Qifti,  flourished  \.i>.  1198  [Colebrooke,  loc. cit., 
note  Vol.  IT,  p  510]. 


EARLY  IDEAS  OF  THEIR  ORIGIN  9 

as  the  Sindum-Indum  with  the  meaning  "perpetuum 
aeternumque."  The  reference 1  in  this  ancient  Arabic 
work  to  Al-Khowarazml  is  worthy  of  note. 

This  Sindhind  is  the  book,  says  Mas'udI,2  which  gives 
all  that  the  Hindus  know  of  the  spheres,  the  stars,  arith- 
metic,3 and  the  other  branches  of  science.  He  mentions 
also  Al-Khowarazml  and  H abash4  as  translators  of  the 
tables  of  the  Sindhind.  Al-Biruni5  refers  to  two  other 
translations  from  a  work  furnished  by  a  Hindu  who 
came  to  Bagdad  as  a  member  of  the  political  mission 
which  Sindh  sent  to  the  caliph  Al-Mansiir,  in  the  year  of 
the  Hejira  154  (a.d.  771). 

The  oldest  work,  in  any  sense  complete,  on  the  history 
of  Arabic  literature  and  history  is  the  Kitdb  al-Fihrist, 
written  in  the  year  987  a.d.,  by  Ihn  Abi  Ya'qub  al-Nadhn. 
It  is  of  fundamental  importance  for  the  history  of  Arabic 
culture.  Of  the  ten  chief  divisions  of  the  work,  the  sev- 
enth demands  attention  in  this  discussion  for  the  reason 
that  its  second  subdivision  treats  of  mathematicians  and 
astronomers.6 


1  "Liber  Artis  Logisticae  a  Mohamado  Ben  Musa  Alkhuarezmila 
exornatus,  qui  ceteros  omnes  brevitate  methodi  ac  facilitate  praestat, 
Indorum  que  in  praeclarissimis  inventis  Lngenium  &  acumen  osten- 
dit."    [Casiri,  loc.  cit.,  p.  427.] 

2  Macoudi,  Le  lime  de  V  avertissement  et  de  la  revision.  Translation 
by  B.  Carra  de  Vaux,  Paris,  1896. 

3  Verifying  the  hypothesis  of  Woepcke,  Propagation,  that  the  Sind- 
hind included  a  treatment  of  arithmetic. 

4  Ahmed  ibn  "Abdallah,  Suter,  Die  MaMiematiker,  etc.,  p.  12. 

5  India,  Vol.  II,  p.  15. 

6  See  H.  Suter,  "Das  Mathematiker-Verzeichniss  im  Fihrist," 
Abhandlungen  zur  Geschichte  der  Mathematik,  Vol.  VI,  Leipzig,  1892. 
For  further  references  to  early  Arabic  writers  the  reader  is  referred 
to  H.  Suter,  Die  Mathematiker  und  Astronomen  der  Araber  und  Hire 
Werke.  Also  "Nachtrage  und  Berichtigungen"  to  the  same  {Abhand- 
lungen, Vol.  XIV,  1902,  pp.  155-180). 


10  THE  HINDU-ARABIC   NUMERALS 

The  first  of  the  Arabic  writers  mentioned  is  Al-Kindi 
(800-870  A.D.),  who  wrote  five  books  on  arithmetic  and 
four  books  on  the  use  of  the  Indian  method  of  reckoning. 
Sened  ibn  'AH,  the  Jew,  who  was  converted  to  Islam  under 
the  caliph  Al-Mamun,  is  also  given  as  the  author  of  a  work 
on  the  Hindu  method  of  reckoning.  Nevertheless,  there 
is  a  possibility  x  that  some  of  the  works  ascribed  to  Sened 
ibn  'All  are  really  works  of  Al-Khowarazmi,  whose  name 
immediately  precedes  his.  However,  it  is  to  be  noted  in 
this  connection  that  Casiri 2  also  mentions  the  same  writer 
as  the  author  of  a  most  celebrated  work  on  arithmetic. 

To  Al-Sufi,  who  died  in  986  a.d.,  is  also  credited  a  large 
work  on  the  same  subject,  and  similar  treatises  by  other 
writers  are  mentioned.  We  are  therefore  forced  to  the 
conclusion  that  the  Arabs  from  the  early  ninth  century 
on  fully  recognized  the  Hindu  origin  of  the  new  numerals. 

Leonard  of  Pisa,  of  whom  we  shall  speak  at  length  in 
the  chapter  on  the  Introduction  of  the  Numerals  into 
Europe,  wrote  his  Liber  Abbaei3  in  1202.  In  this  work 
he  refers  frequently  to  the  nine  Indian  figures,4  thus 
showing  again  the  general  consensus  of  opinion  in  the 
Middle  Ages  that  the  numerals  were  of  Hindu  origin. 

Some  interest  also  attaches  to  the  oldest  documents  on 
arithmetic  in  our  own  language.     One  of  the  earliest 

1  Suter,  loc.  cit.,  note  165,  pp.  62-63. 

2  "  Send  Ben  Ali,  .  .  .  turn  arithmetica  scripta  maxime  celebrata, 
quae  publici  juris  fecit.''    [Loc.  cit.,  p.  440.] 

8  Scritti  di  Leonardo  Pisano,  Vol.  I,  Liber  Abbaei  (1857);  Vol.  II, 
Scritti  (1862);  published  by  Baldassarre  Boncompagni,  Rome.  Also 
Tre  Scritti  Inediti,  and  Intomo  ad  Opere  di  Leonardo  Pisano,  Rome, 
1854. 

4  "  Ubi  ex  mirabili  magisterio  in  arte  per  novem  figuras  indorum 
introductus"  etc.  In  another  place,  as  a  heading  to  a  separate  divi- 
sion, he  writes,  "De  cognitione  novem  figurarum  yndoxum"  etc. 
" Novem  figure  indorum  he  sunt  987654321."   . 


EARLY  IDEAS  OF  THEIR  ORIGIN  11 

treatises  on  algorism  is  a  commentary 1  on  a  set  of 
verses  called  the  Carmen  de  Algorismo,  written  by  Alex- 
ander de  Villa  Dei  (Alexandre  de  Ville-Dieu),  a  Minor- 
ite monk  of  about  1240  a.d.  The  text  of  the  first  few 
lines  is  as  follows  : 

"Hec  algorism'  ars  p'sens  elicit'  in  qua 
Talib;  indor2/  fruim  bis  quinq;  figuris.2 

"  This  boke  is  called  the  boke  of  algorim  or  augrym 
after  lewder  use.  And  this  boke  tretys  of  the  Craft  of 
Nombryng,  the  quych  crafte  is  called  also  Algorym. 
Ther  was  a  kyng  of  Inde  the  quich  heyth  A  Igor  &  he 
made  this  craft.  .  .  .  Algorisms,  hi  the  quych  we  use 
teen  figurys  of  Inde." 

1  See  An  Ancient  English  Algorism,  by  David  Eugene  Smith,  in 
Festschrift  Moritz  Cantor,  Leipzig,  1909.  See  also  Victor  Mortet,  "Le 
plus  ancien  traits'  francais  d'algorisme,"  Bibliotheca  Mathematica,  Vol. 
IX  (3),  pp.  55-64. 

2  These  are  the  two  opening  lines  of  the  Carmen  de  Algorismo  that 
the  anonymous  author  is  explaining.    They  should  read  as  follows : 

Haec  algorismus  ars  praesens  dicitur,  in  qua 
Talibus  Indorum  fruimur  bis  quiuque  figuris. 

"What  follows  is  the  translation. 


CHAPTER  II 
EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE 

While  it  is  generally  conceded  that  the  scientific  de- 
velopment of  astronomy  among  the  Hindus  towards 
the  beginning  of  the  Christian  era  rested  upon  Greek  1 
or  Chinese  2  sources,  yet  their  ancient  literature  testifies 
to  a  high  state  of  civilization,  and  to  a  considerable  ad- 
vance in  sciences,  in  philosophy,  and  along  literary  lines, 
long  before  the  golden  age  of  Greece.  From  the  earliest 
times  even  up  to  the  present  day  the  Hindu  has  been 
wont  to  put  his  thought  into  rhythmic  form.  The  first 
of  this  poetry  —  it  well  deserves  this  name,  being  also 
worthy  from  a  metaphysical  point  of  view  3  —  consists  of 
the  Vedas,  hymns  of  praise  and  poems  of  worship,  col- 
lected during  the  Vedic  period  which  dates  from  approxi- 
mately 2000  B.C.  to  1400  B.C.4  Following  this  work,  or 
possibly  contemporary  with  it,  is  the  Brahmanic  literature, 
which  is  partly  ritualistic  (the  Brahmanas),  and  partly 
philosophical  (the  Upanishads).    Our  especial  interest  is 

1  Thibaut,  Astronomie,  Astrologie  und  Mathematik,  Strassburg, 
1809. 

2  Gustave  Schlegel,  Uranographie  chinoise  ou  preuves  directes  gpie 
V astronomie  primitive  est  originaire  de  la  Chine,  et  qu'elle  a  6U  emprun- 
Ue  par  les  anciens  peuples  occidentaux  a  la  sphere  chinoise ;  ouvrage  ac- 
compagne'  d'un  atlas  celeste  chinois  et  grec,  The  Hague  and  Ley  den, 
1875. 

:!  !•:.  W.  Hopkins,  The  Religions  of  India,  Boston,  1898,  p.  7. 
4  R.  C.  Dutt,  History  of  India,  London,  1906. 

12 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE     13 

in  the  Sutras,  versified  abridgments  of  the  ritual  and  of 
ceremonial  rules,  which  contain  considerable  geometric 
material  used  in  connection  with  altar  construction,  and 
also  numerous  examples  of  rational  numbers  the  sum  of 
whose  squares  is  also  a  square,  i.e.  "  Pythagorean  num- 
bers," although  this  was  long  before  Pythagoras  lived. 
Whitney  *  places  the  whole  of  the  Veda  literature,  includ- 
ing the  Vedas,  the  Brahmanas,  and  the  Sutras,  between 
1500  B.C.  and  800  B.C.,  thus  agreeing  with  Biirk  2  who 
holds  that  the  knowledge  of  the  Pythagorean  theorem  re- 
vealed in  the  Sutras  goes  back  to  the  eighth  century  B.C. 
The  importance  of  the  Sutras  as  showing  an  independ- 
ent origin  of  Hindu  geometry,  contrary  to  the  opinion 
long  held  by  Cantor  3  of  a  Greek  origin,  has  been  repeat- 
edly emphasized  in  recent  literature,4  especially  since 
the  appearance  of  the  important  work  of  Von  Schroeder.5 
Further  fundamental  mathematical  notions  such  as  the 
conception  of  irrationals  and  the  use  of  gnomons,  as  well  as 
the  philosophical  doctrine  of  the  transmigration  of  souls, 

—  all  of  these  having  long  been  attributed  to  the  Greeks, 

—  are  shown  hi  these  works  to  be  native  to  India.    Al- 
though this  discussion  does  not  bear  directly  upon  the 

1  W.  D.  Whitney,  Sanskrit  Grammar,  3d  ed.,  Leipzig,  1896. 

2  "Das  Apastamba-Sulba-Sutra,"  Zeitschrlft  der  deutschen  Morgen- 
landischen  Gesellschaft,  Vol.  LV,  p.  543,  and  Vol.  LVI,  p.  327. 

3  Geschichte  der  Math.,  Vol.  I,  2d  ed.,  p.  595. 

4  L.  von  Schroeder,  Pythagoras  und  die  Inder,  Leipzig,  1884 ;  H. 
Vogt,  "  Haben  die  alten  Inder  den  Pythagoreischen  Lehrsatz  und  das 
Irrationale  gekannt? "  Bibliotheca  Mathematica,  Vol.  VII  (3),  pp.  6-20; 
A.  Biirk,  loc.  cit. ;  Max  Simon,  Geschichte  der  Mathematik  im  Altertum, 
Berlin,  1909,  pp.  137-165 ;  three  Sutras  are  translated  in  part  by 
Thibaut,  Journal  of  the  Asiatic  Society  of  Bengal,  1875,  and  one  ap- 
peared in  The  Pandit,  1875  ;  Beppo  Levi,  "  Osservazioni  e  congetture 
soprala  geometriadegli  indiani,"  Bibliotheca  Mathematica,  Vol.  IX  (3), 
1908,  pp.  97-105. 

5  Loc.  cit.;  also  Indiens  Literatur  und  Cultur,  Leipzig,  1887. 


14  THE  HINDU-ARABIC  NUMERALS 

origin  of  our  numerals,  yet  it  is  highly  pertinent  as  show- 
ing the  aptitude  of  the  Hindu  for  mathematical  and  men- 
tal work,  a  fact  further  attested  by  the  independent 
development  of  the  drama  and  of  epic  and  lyric  poetry. 

It  should  be  stated  definitely  at  the  outset,  however, 
that  we  are  not  at  all  sure  that  the  most  ancient  forms 
of  the  numerals  commonly  known  as  Arabic  had  their 
origin  in  India.  As  will  presently  be  seen,  their  forms 
may  have  been  suggested  by  those  used  in  Egypt,  or  in 
Eastern  Persia,  or  in  China,  or  on  the  plains  of  Mesopo- 
tamia. We  are  quite  in  the  dark  as  to  these  early  steps ; 
but  as  to  their  development  in  India,  the  approximate 
period  of  the  rise  of  their  essential  feature  of  place  value, 
their  introduction  into  the  Arab  civilization,  and  their 
spread  to  the  West,  we  have  more  or  less  definite  infor- 
mation. When,  therefore,  we  consider  the  rise  of  the 
numerals  in  the  land  of  the  Sindhu,1  it  must  be  under- 
stood that  it  is  only  the  large  movement  that  is  meant, 
and  that  there  must  further  be  considered  the  numerous 
possible  sources  outside  of  India  itself  and  long  anterior 
to  the  first  prominent  appearance  of  the  number  symbols. 

No  one  attempts  to  examine  any  detail  in  the  history  of 
ancient  India  without  being  struck  with  the  great  dearth 
of  reliable  material.2  So  little  sympathy  have  the  people 
with  any  save  those  of  their  own  caste  that  a  general  litera- 
ture is  wholly  lacking,  and  it  is  only  in  the  observations 
of  strangers  that  any  all-round  view  of  scientific  progress 
is  to  be  found.    There  is  evidence  that  primary  schools 

1  It  is  generally  agreed  that  the  name  of  the  river  Sindhu,  corrupted 
by  western  peoples  to  Hindhu,  Indos,  Indus,  is  the  root  of  Hindustan 
and  of  India.    Reclus,  Asia,  English  ed.,  Vol.  Ill,  p.  14. 

2  See  the  comments  of  Oppert,  On  the  Original  Inhabitants  of  Bhara- 
tavurm  or  India,  London,  1893,  p.  1. 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE    15 

existed  in  earliest  times,  and  of  the  seventy -two  recognized 
sciences  writing  and  arithmetic  were  the  most  prized.1  In 
the  Vedic  period,  say  from  2000  to  1400  B.C.,  there  was 
the  same  attention  to  astronomy  that  was  found  in  the 
earlier  civilizations  of  Babylon,  China,  and  Egypt,  a  fact  at- 
tested by  the  Vedas  themselves.2  Such  advance  in  science 
presupposes  a  fair  knowledge  of  calculation,  but  of  the 
manner  of  calculating  we  are  quite  ignorant  and  prob- 
ably always  shall  be.  One  of  the  Buddhist  sacred  books, 
the  Lalitavistara,  relates  that  when  the  Bodhisattva3  was 
of  age  to  marry,  the  father  of  Gopa,  his  intended  bride, 
demanded  an  examination  of  the  five  hundred  suitors, 
the  subjects  including  arithmetic,  writing,  the  lute,  and 
archery.  Having  vanquished  his  rivals  in  all  else,  he  is 
matched  against  Arjuna  the  great  arithmetician  and  is 
asked  to  express  numbers  greater  than  100  kotis.4  In 
reply  he  gave  a  scheme  of  number  names  as  high  as  1053, 
adding  that  he  could  proceed  as  far  as  10421,5  all  of  which 
suggests  the  system  of  Archimedes  and  the  unsettled 
question  of  the  indebtedness  of  the  West  to  the  East  in 
the  realm  of  ancient  mathematics.6    Sir  Edwin  Arnold, 

1  A.  Hillebrandt,  Alt-Indicn,  Breslau,  1899,  p.  111.  Fragmentary 
records  relate  that  Kharavela,  king  of  Kalinga,  learned  as  a  boy  lekha 
(writing),  ganana  (reckoning),  and  rupa  (arithmetic  applied  to  mone- 
tary affairs  and  mensuration),  probably  in  the  5th  century  n.c. 
[Biihler,  Indische  Palaeographie,  Strassburg,  1896,  p.  5.] 

2  R.  C.  Dutt,  A  History  of  Civilization  in  Ancient  India,  London, 
1893,  Vol.  I,  p.  174. 

3  The  Buddha.  The  date  of  his  birth  is  uncertain.  Sir  Edwin  Ar- 
nold put  it  c.  620  b.c. 

.  *  I.e.  100-107. 

5  There  is  some  uncertainty  about  this  limit. 

6  This  problem  deserves  more  study  than  has  yet  been  given  it.  A 
beginning  may  be  made  with  Comte  Goblet  d'Alviella,  Ce  que  Vlnde 
doit  a  la  Grece,  Paris,  1897,  and  H.  G.  Keene's  review,  "  The  Greeks  in 
India,"  in  the  Calcutta  Review,  Vol,  CXIV,  1902,  p.  1.    See  also  F. 


16  THE  HINDU-ARABIC  NUMERALS 

hi  The  Light  of  Asia,  does  not  mention  this  part  of  the 

contest,  but  he  speaks  of  Buddha's  training  at  the  hands 

of  the  learned  Visvamitra : 

"  And  Viswamitra  said,  '  It  is  enough, 

Let  us  to  numbers.    After  me  repeat 

Your  numeration  till  we  reach  the  lakh,1 

One,  two,  three,  four,  to  ten,  and  then  by  tens 

To  hundreds,  thousands.'    After  him  the  child 

Named  digits,  decads,  centuries,  nor  paused, 

The  round  lakh  reached,  but  softly  murmured  on, 

Then  comes  the  koti,  nahut,  niunahut, 

Khamba,  viskhamba,  abab,  attata, 

To  kumuds,  gundhikas,  and  utpalas, 

By  pundarikas  into  padumas, 

Which  last  is  how  you  count  the  utmost  grains 

Of  Hastagiri  ground  to  finest  dust ;  2 

But  beyond  that  a  numeration  is, 

The  Katha,  used  to  count  the  stars  of  night, 

The  Koti-Katha,  for  the  ocean  drops ; 

Ingga,  the  calculus  of  circulars ; 

Sarvanikchepa,  by  the  which  you  deal 

With  all  the  sands  of  Gunga,  till  we  come 

To  Antah-Kalpas,  where  the  unit  is 

The  sands  of  the  ten  crore  Gungas.    If  one  seeks 

More  comprehensive  scale,  th'  arithmic  mounts 

By  the  Asankya,  which  is  the  tale 

Of  all  the  drops  that  in  ten  thousand  years 

Would  fall  on  all  the  worlds  by  daily  rain; 

Thence  unto  Maha  Kalpas,  by  the  which 

The  gods  compute  their  future  and  their  past.'  " 

Woepcke,  Propagation,  p.  253;  G.  R.  Kaye,  loc.  cit.,  p.  47-r>seq.,  and 
"The  Source  of  Hindu  Mathematics,"  Journal  of  the  Royal  Asiatic 
Society,  July,  1910,  pp.  749-700;  G.  Thibaut,  Astronomic,  Astroloyic 
und  Mathematik,  pp.  43-50  and  76-79.  It  will  be  discussed  more  fully 
in  Chapter  VI. 

1  I.e.  to  100,000.  The  lakh  is  still  the  common  large  unit  in  India, 
like  the  myriad  in  ancient  Greece  and  the  million  in  the  West. 

2  This  again  suggests  the  Psammites,  or  De  harenae  numero  as  it  is 
called  in  the  1544  edition  of  the  Opera  of  Archimedes,  a  work  in  which 
the  great  Syracnsan  proposes  to  show  to  the  king  "  by  geometric  proofs 
which  you  can  follow,  that  the  numbers  which  have  been  named  by 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE    17 

Thereupon  Visvamitra  Acarya  1  expresses  his  approval 
of  the  task,  and  asks  to  hear  the  "  measure  of  the  line  " 
as  far  as  yojana,  the  longest  measure  bearing  name.  This 
given,  Buddha  adds  : 

.  .  .  "  '  And  master  !  if  it  please, 
I  shall  recite  how  many  sun-motes  lie 
From  end  to  end  within  a  yojana.' 
Thereat,  with  instant  skill,  the  little  prince 
Pronounced  the  total  of  the  atoms  true. 
But  Viswamitra  heard  it  on  his  face 
Prostrate  before  the  boy ;  '  For  thou,'  he  cried, 
'  Art  Teacher  of  thy  teachers  —  thou,  not  I, 
Art  Guru.'  " 

It  is  needless  to  say  that  this  is  far  from  being  history. 
And  yet  it  puts  in  charming  rhythm  only  what  the  ancient 
Lalitavistara  relates  of  the  number-series  of  the  Buddha's 
time.  While  it  extends  beyond  all  reason,  nevertheless 
it  reveals  a  condition  that  would  have  been  impossible 
unless  arithmetic  had  attained  a  considerable  decree  of 
advancement. 

To  this  pre-Christian  period  belong  also  the  Veddhgas, 
or  "  limbs  for  supporting  the  Veda,"  part  of  that  great 
branch  of  Hindu  literature  known  as  Smriti  (recollec- 
tion), that  which  was  to  be  handed  down  by  tradition. 
Of  these  the  sixth  is  known  as  Jyotim  (astronomy),  a 
short  treatise  of  only  thirty -six  verses,  written  not  earlier 
than  300  B.C.,  and  affording  us  some  knowledge  of  the 
extent  of  number  work   in   that  period.2    The  Hindus 

us  .  .  .  are  sufficient  to  exceed  not  only  the  number  of  a  sand-heap  as 
large  as  the  whole  earth,  but  one  as  large  as  the  universe."  For  a 
list  of  early  editions  of  this  work  see  D.  E.  Smith,  Eara  Arithmetical 
Boston,  1909,  p.  227.  i  I.e.  the  Wise. 

2  Sir  Monier  Monier- Williams,  Indian  Wisdom,  4th  ed.,  London, 
1893,  pp.  144,  177.  See  also  J:  C.  Marshman,  Abridgment  of  the  History 
of  India,  London,  1893,  p.  2. 


18  THE  HINDU-ARABIC  NUMERALS 

also  speak  of  eighteen  ancient  Siddhantas  or  astronomical 
works,  which,  though  mostly  lost,  confirm  this  evidence.1. 

As  to  authentic  histories,  however,  there  exist  in  India 
none  relating  to  the  period  before  the  Mohammedan  era 
(622  a.d.).  About  all  that  we  know  of  the  earlier  civi- 
lization is  what  we  glean  from  the  two  great  epics,  the 
Mahabharata  2  and  the  Ramayana,  from  coins,  and  from 
a  few  inscriptions.3 

It  is  with  this  unsatisfactory  material,  then,  that  we 
have  to  deal  in  searching  for  the  early  history  of  the 
Hindu-Arabic  numerals,  and  the  fact  that  many  unsolved 
problems  exist  and  will  continue  to  exist  is  no  longer 
strange  when  we  consider  the  conditions.  It  is  rather 
surprising  that  so  much  has  been  discovered  within  a 
century,  than  that  we  are  so  uncertain  as  to  origins  and 
dates  and  the  early  spread  of  the  system.  The  probabil- 
ity being  that  writing  was  not  introduced  into  India 
before  the  close  of  the  fourth  century  B.C.,  and  literature 
existing  only  in  spoken  form  prior  to  that  period,4  the 
number  work  was  doubtless  that  of  all  primitive  peoples, 
palpable,  merely  a  matter  of  placing  sticks  or  cowries  or 
pebbles  on  the  ground,  of  marking  a  sand-covered  board, 
or  of  cutting  notches  or  tying  cords  as  is  still  done  in 
parts  of  Southern  India  to-day.5 

1  For  a  list  and  for  some  description  of  these  works  see  R.  C.  Dutt, 
A  History  of  Civilization  in  Ancient  India,  Vol.  II,  p.  121. 

2  Professor  Ramkrishna  Gopal  Bhandarkar  fixes  the  date  as  the 
fifth  century  b.c.  ["Consideration  of  the  Date  of  the  Mahabharata." 
in  the  Journal  of  the  Bombay  Branch  of  the  R.  A.  Soc,  Bombay,  1873, 
Vol.  X,  p.  2.] 

:i  Marshman,  loc.  cit.,  p.  2. 

4  A.  C.  Burnell,  Mouth  Indian  Pakt'oaraphy,  2d  ed.,  London,  1878, 
p.  1,  seq. 

5  This  extensive  subject  of  palpable  arithmetic,  essentially  the 
history  of  the  abacus,  deserves  to  be  treated  in  a  work  by  itself. 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE     19 

The  early  Hindu  numerals 1  may  be  classified  into 
three  great  groups,  (1)  the  Kharosthi,  (2)  the  Brahmi, 
and  (3)  the  word  and  letter  forms ;  and  these  will  be 
considered  in  order. 

The  Kharosthi  numerals  are  found  in  inscriptions  for- 
merly known  as  Bactrian,  Indo-Bactrian,  and  Aryan, 
and  appearing  in  ancient  Gandhara,  now  eastern  Afghan- 
istan and  northern  Punjab.  The  alphabet  of  the  language 
is  found  in  inscriptions  dating  from  the  fourth  century 
B.C.  to  the  third  century  A.D.,  and  from  the  fact  that 
the  words  are  written  from  right  to  left  it  is  assumed  to 
be  of  Semitic  origin.  No  numerals,  however,  have  been 
found  in  the  earliest  of  these  inscriptions,  number-names 
probably  having  been  written  out  in  words  as  was  the 
custom  with  many  ancient  peoples.  Not  until  the  time 
of  the  powerful  King  Asoka,  in  the  third  century  B.C., 
do  numerals  appear  in  any  inscriptions  thus  far  discov- 
ered ;  and  then  only  in  the  primitive  form  of  marks,  quite 
as  they  would  be  found  hi  Egypt,  Greece,  Rome,  or  in 

1  The  following  are  the  leading  sources  of  information  upon  this 
subject :  G.  Buhler,  Indische  Palaeographie,  particularly  chap,  vi ; 
A.  C.  Burnell,  South  Indian  Palaeography,  2ded.,  London,  1878,  where 
tables  of  the  various  Indian  numerals  are  given  in  Plate  XXIII ;  E.  C. 
Bayley,  "  On  the  Genealogy  of  Modern  Numerals,"  Journal  of  the  Eoyal 
Asiatic  Society,  Vol.  XIV,  part  3,  and  Vol.  XV,  part  1,  and  reprint, 
London,  1882 ;  I.  Taylor,  in  The  Academy,  January  28,  1882,  with  a 
repetition  of  his  argument  in  his  work  The  Alphabet,  London,  1883, 
Vol.  II,  p.  265,  based  on  Bayley  ;  G.  R.  Kaye,  loc.  cit.,  in  some  respects 
one  of  the  most  critical  articles  thus  far  published;  J.  C.  Fleet, 
Corpus  inscriptionum  Indicarum,  London,  1888,  Vol.  Ill,  with  fac- 
similes of  many  Indian  inscriptions,  and  Indian  Epigraphy,  Oxford, 
1907,  reprinted  from  the  Imperial  Gazetteer  of  India,  Vol.  II,  pp.  1-88, 
1907;  G.  Thibaut,  loc.  cit.,  Astronomic  etc. ;  R.Caldwell,  Comparative 
Grammar  of  the  Dravidian  Languages,  London,  1856,  p.  262  seq.;  and 
Epigraphia  Indica  (official  publication  of  the  government  of  India), 
Vols.  I-IX.  Another  work  of  Buhler' s,  On  the  Origin  of  the  Indian 
Brahma  Alphabet,  is  also  of  value. 


20  THE  HINDU-ARABIC  NUMERALS 

various  other  parts  of  the  world.  These  Asoka x  inscrip- 
tions, some  thirty  in  all,  are  found  in  widely  separated 
parts  of  India,  often  on  columns,  and  are  in  the  various 
vernaculars  that  were  familiar  to  the  people.  Two  are  in 
the  Kharosthi  characters,  and  the  rest  in  some  form  of 
Brahml.  In  the  Kharosthi  inscriptions  only  four  numer- 
als have  been  found,  and  these  are  merely  vertical  marks 
for  one,  two,  four,  and  five,  thus : 

I  II  MM  Mill 

In  the  so-called  Saka  inscriptions,  possibly  of  the  first 
century  B.C.,  more  numerals  are  found,  and  in  more 
highly  developed  form,  the  right-to-left  system  appearing, 
together  with  evidences  of  three  different  scales  of  count- 
ing, —  four,  ten,  and  twenty.  The  numerals  of  this 
period  are  as  follows : 

12345  6  8  10 

'     II    HI     X    IX    II*    XX     ? 

3  933   333    9313    Xl     til 

20  50  GO  70  100  200 

There  are  several  noteworthy  points  to  be  observed  in 
studying  this  system.  In  the  first  place,  it  is  probably  not 
as  early  as  that  shown  in  the  Nana  Ghat  forms  hereafter 
given,  although  the  inscriptions  themselves  at  Nana 
Ghat  are  later  than  those  of  the  Asoka  period.    The 

1  The  earliest  work  on  the  subject  was  by  James  Prinsep,  "On  the 
Inscriptions  of  Piyadasi  or  Asoka,"  etc.,  Journal  of  the  Asiatic  Society 
of  Bengal,  1838,  following  a  preliminary  suggestion  in  the  same  journal 
in  1837.  See  also  "Asoka  Notes,"  by  V.  A.  Smith,  The  Indian  An- 
tiquary, Vol.  XXXVII,  1908,  p.  24seq.,  Vol.  XXXVIII,  pp.  151-159, 
June,  1909 ;  The  Early  History  of  India,  2d  ed.,  oxford.  1908,  p.  154; 
J.  F.  Fleet,  "The  Last  Words  of  Asoka,"  Journal  of  the  Royal  Asiatic 
Society,  October,  1909,  pp.  981-1016;  E.  Senart,  Les  inscriptions  de 
Piyadasi,  2  vols.,  Paris,  1887. 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE    21 

four  is  to  this  system  what  the  X  was  to  the  Roman, 
probably  a  canceling  of  three  marks  as  a  workman  does 
to-day  for  five,  or  a  laying  of  one  stick  across  three  others. 
The  ten  has  never  been  satisfactorily  explained.  It  is 
similar  to  the  A  of  the  Kharosthi  alphabet,  but  we  have 
no  knowledge  as  to  why  it  was  chosen.  The  twenty  is 
evidently  a  ligature  of  two  tens,  and  this  in  turn  sug- 
gested a  kind  of  radix,  so  that  ninety  was  probably  writ- 
ten in  a  way  reminding  one  of  the  quatre-vingt-dix  of 
the  French.  The  hundred  is  unexplained,  although  it 
resembles  the  letter  ta  or  tra  of  the  Brahmi  alphabet  with 
1  before  (to  the  right  of)  it.  The  two  hundred  is  only 
a  variant  of  the  symbol  for  hundred,  with  two  vertical 
marks.1 

This  system  has  many  points  of  similarity  with  the 
Nabatean  numerals2  in  use  in  the  first  centuries  of  the 
Christian  era.  The  cross  is  here  used  for  four,  and  the 
Kharosthi  form  is  employed  for  twenty.  In  addition  to 
this  there  is  a  trace  of  an  analogous  use  of  a  scale  of 
twenty.  While  the  symbol  for  100  is  quite  different,  the 
method  of  forming  the  other  hundreds  is  the  same.  The 
correspondence  seems  to  be  too  marked  to  be  wholly 
accidental. 

It  is  not  in  the  Kharosthi  numerals,  therefore,  that  we 
can  hope  to  find  the  origin  of  those  used  by  us,  and  we 
turn  to  the  second  of  the  Indian  types,  the  Brahmi  char- 
acters. The  alphabet  attributed  to  Brahma  is  the  oldest  of 
the  several  known  in  India,  and  was  used  from  the  earliest 
historic  times.    There  are  various  theories  of  its  origin, 

1  For  a  discussion  of  the  minor  details  of  this  system,  see  Biihler, 
loc.  cit.,  p.  73. 

2  Julius  Euting,  Nabataische  Inschriften  aus  Arabien,  Berlin,  1885, 
pp.  96-97,  with  a  table  of  numerals. 


22  THE  HINDU-ARABIC  NUMERALS 

none  of  which  has  as  yet  any  wide  acceptance,1  although 
the  problem  offers  hope  of  solution  in  due  time.  The 
numerals  are  not  as  old  as  the  alphabet,  or  at  least  they 
have  not  as  yet  been  found  in  inscriptions  earlier  than 
those  in  which  the  edicts  of  Asoka  appear,  some  of  these 
having  been  incised  in  Brahnu  as  well  as  KharosthL  As 
already  stated,  the  older  writers  probably  wrote  the  num- 
bers in  words,  as  seems  to  have  been  the  case  in  the 
earliest  Pali  writings  of  Ceylon.2 

The  following  numerals  are,  as  far  as  known,  the 
only  ones  to  appear  in  the  Asoka  edicts : 3 

\\\  +  <lf  S  0  A   yr  1 

12        4  6  50        50       200  200  200 

These  fragments  from  the  third  century  B.C.,  crude  and 
unsatisfactory  as  they  are,  are  the  undoubted  early  forms 
from  which  our  present  system  developed.  They  next 
appear  in  the  second  century  B.C.  in  some  inscriptions  in 
the  cave  on  the  top  of  the  Nana  Ghat  hill,  about  seventy- 
five  miles  from  Poona  in  central  India.  These  inscrip- 
tions may  be  memorials  of  the  early  Andhra  dynasty  of 
southern  India,  but  their  chief  interest  lies  in  the  numer- 
als winch  they  contain. 

The  cave  was  made  as  a  resting-place  for  travelers  as- 
cending the  hill,  which  lies  on  the  road  from  Kalyana  to 
Junar.    It  seems  to  have  been  cut  out  by  a  descendant 

1  For  the  five  principal  theories  see  Biihler,  loc.  cit.,  p.  10. 

2  Bayley,  loc.  cit.,  reprint  p.  3. 

3  Biihler,  loc.  cit.;  Epigraphia  Indica,  Vol.  Ill,  p.  134 ;  Indian  An- 
tiquary, Vol.  VI,  p.  155  seq.,  and  Vol.  X,  p.  107. 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE     23 

of  King  Satavahana,1  for  inside  the  wall  opposite  the  en- 
trance are  representations  of  the  members  of  his  family, 
much  defaced,  but  with  the  names  still  legible.  It  would 
seem  that  the  excavation  was  made  by  order  of  a  king 
named  Vedisiri,  and  "  the  inscription  contains  a  list  of 
gifts  made  on  the  occasion  of  the  performance  of  several 
yagnas  or  religious  sacrifices,"  and  numerals  are  to  be 
seen  in  no  less  than  thirty  places.2 

There  is  considerable  dispute  as  to  what  numerals  are 
really  found  in  these  inscriptions,  owing  to  the  difficulty 
of  deciphering  them  ;  but  the  following,  which  have  been 
copied  from  a  rubbing,  are  probably  number  forms :  3 

_=^P^f-      <f    1    p'acerar 

12  4  6  7  9  10        10         10 

o  H   a>  W)-\  KH  iff 

20    GO     80      100    100   100       200         400 

W  T  T    Tf  Ty  Tor    T° 

700      1000        4000      G000      10,000      20,000 

The  inscription  itself,  so  important  as  containing  the 
earliest  considerable  Hindu  numeral  system,  connected 
with  our  own,  is  of  sufficient  interest  to  warrant  repro- 
ducing part  of  it  hi  facsimile,  as  is  done  on  page  24. 

1  Pandit  Bhagavanlal  Indraji,  "  On  Ancient  Nagari  Numeration ; 
from  an  Inscription  at  Naneghat,"  Journal  of  the  Bombay  Branch  of  the 
Royal  Asiatic  Society,  1876,  Vol.  XII,  p.  404. 

2  lb.,  p.  405.  He  gives  also  a  plate  and  an  interpretation  of  each 
numeral. 

3  These  may  be  compared  with  Bidder's  drawings,  loc.  cit. ;  with 
Bayley,  loc.  cit.,  p.  337  and  plates  ;  and  with  Bayley's  article  in  the 
Encyclopaedia  Britannica,  9th  ed.,  art.  "Numerals." 


24 


THE  HINDU-ARABIC  NUMERALS 


flanacjhat    Inscriptions 


itffl 


The  next  very  noteworthy  evidence  of  the  numerals, 
and  this  quite  complete  as  will  be  seen,  is  found  in  cer- 
tain other  cave  inscriptions  dating  back  to  the  first  or 
second  century  A.D.  In  these,  the  Nasik  1  cave  inscrip- 
tions, the  forms  are  as  follows: 

1  2  3  4  5  C         7  8  9 

10  10  20  40  70  100         200  500 

5f     f    f    T     P    V 

1000     2000     3000     4000       8000     70,000 

From  this  time  on,  until  the  dechnal  system  finally 
adopted  the  first  nine  characters  and  replaced  the  rest  of 
the  Brahnri  notation  by  adding  the  zero,  the  progress  of 
these  forms  is  well  marked.  It  is  therefore  well  to  present 

1  E.  Senart,  "The  Inscriptions  in  the  Caves  at  Nasik,"  Epigraphia 
Indira,  Vol.  VIII,  pp.  59-96 ;  "The  Inscriptions  in  the  Cave  at  Karle," 
Epigraphia  Iudica,  Vol.  VII,  pp.  47-74;  Buhler,  Palaeographie,  Tafel 
IX. 


G 

AIM 

-\ 

& 

»KT 

X 

IT? 

EARLY  HINDU  FOEMS  WITH  NO  PLACE  VALUE    25 

Table  showing  the  Progress  of  Number  Forms 

in  India 

NUMERALS  1    2     3     4     5     6    7     8     9    10  20    30  40    50    GO  70  80   90  1002001000 
»ASoka         |   \\   Will/ 

»Saka  |   ||    WXIXIIX      XX        ?    3 

e  Asoka         I    J  |        -f-       (4> 
*NagarI      —  =       ^       Cp  "}        PoTO 
^eNasik         _-  =  ¥    I1    £   7    <7  ?  OC  @ 
i  Ksatrapa  -  =  =%  f   f  0   3  J  «"  6  />T  *   jy^Offi)^ 

•  Valhabl      N  =  £         3f   f  ^F^©^^^^       T 
J  Nepal  s5i  1  £  ft         T^Q^M  °         ** 

kKalinga         «?£  '     Ji    #  ?  Jf        tf    O        ,*  CO        *3 

i  Vakataka      •  fn  &j     \  Ui     Q 

«■  KharosthI  numerals,  Asoka  inscriptions,  c.  250  b.c.  Senart,  Notes 
d'epigraphie  indienne.    Given  by  Biihler,  loc.  cit.,  Tafel  I. 

b  Same,  Saka  inscriptions,  probably  of  the  firs£  century  b.c. 
Senart,  loc.  cit. ;  Biihler,  loc.  cit. 

c  Brahml  numerals,  Asoka  inscriptions,  c.  250  b.c  Indian  Anti- 
quary, Vol.  VI,  p.  155  seq. 

d  Same,  Nana  Ghat  inscriptions,  c.  150  b.c.  Bhagavanlal  Indraft 
Ou  Ancient  Nagari  Numeration,  loc.  cit.  Copied  from  a  squeeze  of 
the  original. 

c  Same,  Nasik  inscription,  c.IOOb.c  Burgess,  Archeological  Survey 
Report,  Western  India  ;  Senart,  Epigraphia  Indica,  Vol.  VII,  pp.  47- 
79,  and  Vol.  VIII,  pp.  59-06. 

f  Ksatrapa  coins,  c.  200  a. i>.  Journal  of  the  Eoyal  Asiatic  Society, 
1890,  p.  639. 

s  Kusana  inscriptions,  c.  150  a. d.  Epigraphia  Indica,  Vol.  I,  p.  381, 
and  Vol.  II,  p.  201. 

h  Gupta  Inscriptions,  c.  300  a. n.  to  450  a. n.   Fleet,  loc.  cit.,  Vol.  III. 

i  Valhabi,  c.  600  a.d.    Corpus,  Vol.  III. 

J  BendalPs  Table  of  Numerals,  in  Cat.  Sansk.  Budd.  MSS.,  British 
Museum. 

k  Indian  Antiquary,  Vol.  XIII,  120;  Epigraphia  Indira,  Vol.  Ill, 
127  ff.  !  Fleet,  loc.  cit. 

[Most  of  these  numerals  are  given  by  Biihler,  loc.  cit.,  Tafel  IX.] 


26  THE  HINDU-ARABIC  NUMERALS 

synoptically  the  best-known  specimens  that  have  .come 
down  to  us,  and  this  is  done  in  the  table  on  page  25.1 

With  respect  to  these  numerals  it  should  first  be  noted 
that  no  zero  appears  in  the  table,  and  as  a  matter  of  fact 
none  existed  in  any  of  the  cases  cited.  It  was  therefore 
impossible  to  have  any  place  value,  and  the  numbers  like 
twenty,  thirty,  and  other  multiples  of  ten,  one  hundred, 
and  so  on,  required  separate  symbols  except  where  they 
were  written  out  in  words.  The  ancient  Hindus  had  no 
less  than  twenty  of  these  symbols,2  a  number  that  was 
afterward  greatly  increased.  The  following  are  examples 
of  then  method  of  indicating  certain  numbers  between 
one  hundred  and  one  thousand : 


3   yj/y      for  174  *HO<3- 

6  vh>i  for2e9  6*>i7)~  fi 


for  191 


for  35G 


1  See  Fleet,  loc.  cit.  See  also  T.  Benfey,  Sanskrit  Grammar,  Lon- 
don, 1863,  p.  217  ;  M.  R.  Kale,  Higher  Sanskrit  Grammar,  2d  ed.,  Bom- 
bay, 1898,  p.  110,  and  other  authorities  as  cited. 

2  Bayley,  loc.  cit.,  p.  335. 

3  From  a  copper  plate  of  493  a.b.,  found  at  Karltalai,  Central 
India.  [Fleet,  loc.  cit.,  Plate  XVI.]  It  should  be  stated,  however,  that 
many  of  these  copper  plates,  being  deeds  of  property,  have  forged 
dates  so  as  to  give  the  appearance  of  antiquity  of  title.  On  the  other 
hand,  as  Colebrooke  long  ago  pointed  out,  a  successful  forgery  has 
to  imitate  the  writing  of  the  period  in  question,  so  that  it  becomes 
evidence  well  worth  considering,  as  shown  in  Chapter  III. 

4  From  a  copper  plate  of  510  a.d.,  found  at  Majhgawain,  Central 
India.    [Fleet,  loc.  cit.,  Plate  XIV.] 

5  From  an  inscription  of  588  a.d.,  found  at  Bodh-Gaya,  Bengal 
Presidency.    [Fleet,  loc.  cit.,  Plate  XXIV.] 

6  From  a  copper  plate  of  571  a.d.,  found  at  Maliya,  Bombay  Presi- 
dency.   [Fleet,  loc.  cit.,  Plate  XXIV.] 

7  From  a  Bijayagadh  pillar  inscription  of  372  a.d.  [Fleet,  loc.  cit., 
Plate  XXXVI,  C] 

8  From  a  copper  plate  of  434  a.d.    [Indian  Antiquary,  Vol.  I,  p.  00.] 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE    27 

To  these  may  be  added  the  following  numerals  below 
one  hundred,  similar  to  those  in  the  table : 

QQ  »      for  DO  Cj  2      for  70 

We  have  thus  far  spoken  of  the  Kharosthl  and  Brahmi 
numerals,  and  it  remains  to  mention  the  third  type,  the 
word  and  letter  forms.  These  are,  however,  so  closely 
connected  with  the  perfecting  of  the  system  by  the  inven- 
tion of  the  zero  that  they  are  more  appropriately  consid- 
ered in  the  next  chapter,  particularly  as  they  have  little 
relation  to  the  problem  of  the  origin  of  the  forms  known 
as  the  Arabic. 

Having  now  examined  types  of  the  early  forms  it  is 
appropriate  to  turn  our  attention  to  the  question  of  then- 
origin.  As  to  the  first  three  there  is  no  question.  The 
I  or  —  is  simply  one  stroke,  or  one  stick  laid  down  by 
the  computer.  The  1 1,  or  —  represents  two  strokes  or 
two  sticks,  and  so  for  the  III  and  E  .  From  some  primi- 
tive 1 1  came  the  two  of  Egypt,  of  Rome,  of  early  Greece, 
and  of  various  other  civilizations.  It  appears  in  the 
three  Egyptian  numeral  systems  in  the  following  forms : 

Hieroglyphic  |  I 
Hieratic  (,j 

Demotic  M  W 

The  last  of  these  is  merely  a  cursive  form  as  in  the 
Arabic  l\  which  becomes  our  2  if  tipped  through  a 
right  angle.    From  some  primitive  —  came  the  Chinese 

1  Gadhwa  inscription,  c.  417  a.d.    [Fleet,  loc.  cit.,  Plate  IV,  D.] 

2  Karitalal  plate  of  493  a.d.,  referred  to  above. 


28  THE  HINDU-ARABIC  NUMERALS 

symbol,  which  is  practically  identical  with  the  symbols 
found  commonly  in  India  from  150  B.C.  to  700  a.d.  In 
the  cursive  form  it  becomes  z,  and  this  was  frequently 
used  for  two  in  Germany  until  the  18th  century.  It 
finally  went  into  the  modern  form  2,  and  the  =  in  the 
same  way  became  our  3. 

There  is,  however,  considerable  ground  for  interesting 
speculation  with  respect  to  these  first  three  numerals. 
The  earliest  Hindu  forms  were  perpendicular.  In  the 
Nana  Ghat  inscriptions  they  are  vertical.  But  long  before 
either  the  Asoka  or  the  Nana  Ghat  inscriptions  the  Chi- 
nese were  using  the  horizontal  forms  for  the  first  three 
numerals,  but  a  vertical  arrangement  for  four.1  Now 
where  did  China  get  these  forms  ?  Surely  not  from 
India,  for  she  had  them,  as  her  monuments  and  litera- 
ture 2  show,  long  before  the  Hindus  knew  them.  The 
tradition  is  that  China  brought  her  civilization  around 
the  north  of  Tibet,  from  Mongolia,  the  primitive  habitat 
being  Mesopotamia,  or  possibly  the  oases  of  Turkestan. 
Now  what  numerals  did  Mesopotamia  use  ?  The  Baby- 
lonian system,  simple  in  its  general  principles  but  very 
complicated  in  many  of  its  details,  is  now  well  known.3 
In  particular,  one,  two,  and  three  were  represented  by 
vertical  arrow-heads.    Why,  then,  did  the  Chinese  write 


1  It  seems  evident  that  the  Chinese  four,  curiously  enough  called 
"  eight  in  the  mouth,"  is  only  a  cursive  1 1 1 1. 

2  Chalfont,  F.  H.,  Memoirs  of  the  Carnegie  Museum,  Vol.  IV,  no.  1  ; 
J.  Hager,  An  Explanation  of  the  Elementary  Characters  of  the  Chinese, 
London,  1801. 

3  H.  V.  Hilprecht,  Mathematical,  Metrolocjical  and  Chronological 
Tablets  from  the  Temple  Library  at  Nippur,  Vol.  XX,  part  I.  of  Scries 
A,  Cuneiform  Texts  Published  by  the  Babylonian  Expedition  of  the 
University  of  Pennsylvania,  190*5 ;  A.  Eisenlohr,  Eiu  altbabylonischer 
Felderplan,  Leipzig,  1900:  Maspero,  Dawn  of  Civilization,  p.  773. 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE    29 

theirs  horizontally  ?  The  problem  now  takes  a  new  inter- 
est when  we  find  that  these  Babylonian  forms  were  not 
the  primitive  ones  of  this  region,  but  that  the  early 
Sumerian  forms  were  horizontal.1 

What  interpretation  shall  be  given  to  these  facts  ? 
Shall  we  say  that  it  was  mere  accident  that  one  people 
wrote  "  one  "  vertically  and  that  another  wrote  it  horizon- 
tally ?  This  may  be  the  case  ;  but  it  may  also  be  the 
case  that  the  tribal  migrations  that  ended  in  the  Mongol 
invasion  of  China  started  from  the  Euphrates  while  yet 
the  Sumerian  civilization  was  prominent,  or  from  some 
common  source  in  Turkestan,  and  that  they  carried  to 
the  East  the  primitive  numerals  of  their  ancient  home, 
the  first  three,  these  being  all  that  the  people  as  a  whole 
knew  or  needed.  It  is  equally  possible  that  these  three 
horizontal  forms  represent  primitive  stick-lay ing,  the  most 
natural  position  of  a  stick  placed  in  front  of  a  calculator 
being  the  horizontal  one.  When,  however,  the  cuneiform 
writing  developed  more  fully,  the  vertical  form  may  have 
been  proved  the  easier  to  make,  so  that  by  the  time  the 
migrations  to  the  West  began  these  were  hi  use,  and 
from  them  came  the  upright  forms  of  Egypt,  Greece, 
Rome,  and  other  Mediterranean  lands,  and  those  of 
Asoka's  time  in  India.  After  Asoka,  and  perhaps  among 
the  merchants  of  earlier  centuries,  the  horizontal  forms 
may  have  come  down  into  India  from  China,  thus  giving 
those  of  the  Nana,  Ghat  cave  and  of  later  inscriptions.  This 
is  hi  the  realm  of  speculation,  but  it  is  not  improbable  that 
further  epigraphical  studies  may  confirm  the  hypothesis. 

1  Sir  H.  H.  Howard,  "On  the  Earliest  Inscriptions  from  Chaldea,1' 
Proceedings  of  the  Society  of  B iblical  Archaeology,  XXI,  p.  301,  London, 
1899. 


30  THE  HINDU-ARABIC  NUMERALS 

As  to  the  numerals  above  three  there  have  been  very 
many  conjectures.  The  figure  one  of  the  Demotic  looks 
like  the  one  of  the  Sanskrit,  the  two  (reversed)  like  that  of 
the  Arabic,  the  four  has  some  resemblance  to  that  in  the 
Nasik  caves,  the  five  (reversed)  to  that  on  the  Ksatrapa 
coins,  the  nine  to  that  of  the  Kusana  inscriptions,  and 
other  points  of  similarity  have  been  imagined.  Some 
have  traced  resemblance  between  the  Hieratic  five  and 
seven  and  those  of  the  Indian  inscriptions.  There  have 
not,  therefore,  been  wanting  those  who  asserted  an  Egyp- 
tian origin  for  these  numerals.1  There  has  already  been 
mentioned  the  fact  that  the  Kharosthi  numerals  were 
formerly  known  as  Bactrian,  Indo-Bactrian,  and  Aryan. 
Cunningham2  was  the  first  to  suggest  that  these  nu- 
merals were  derived  from  the  alphabet  of  the  Bactrian 
civilization  of  Eastern  Persia,  perhaps  a  thousand  years 
before  our  era,  and  in  this  he  was  supported  by  the 
scholarly  work  of  Sir  E.  Clive  Bayley,3  who  in  turn 
was  followed  by  Canon  Taylor.4  The  resemblance  has 
not  proved  convincing,  however,  and  Bayley's  drawings 

1  For  a  bibliography  of  the  principal  hypotheses  of  this  nature  see 
Biihler,  loc.  cit.,  p.  77.  Biihler  (p.  78)  feels  that  of  all  these  hypotheses 
that  which  connects  the  Brahmi  with  the  Egyptian  numerals  is  the 
most  plausible,  although  he  does  not  adduce  any  convincing  proof. 
Th.  Henri  Martin,  "Les  signes  nume>aux  et  l'arithm^tique  chez  les 
peuples  de  l'anti quite"  et  du  moyen  age"  (being  an  examination  of 
Cantor's  Mathematische  Beitrdye  zum  Culturlebcn  der  Volher),  Annul i  di 
matematica pura  ed  applicata,  Vol.V,  Rome,  18G4,  pp.  8,  70.  Also,  sam9 
author,  "  Recherches  nouvelles  sur  l'origine  de  notre  systeme  de  nu- 
meration £crite,"  Revue  ArcMologlque,  1857,  pp.  36,  55.  See  also  the 
tables  given  later  in  this  work. 

2  Journal  of  the  Royal  Asiatic  Society,  Bombay  Branch,  Vol.  XXIII. 

3  Loc.  cit.,  reprint,  Part  I,  pp.  12,  17.  Bayley's  deductions  are 
generally  regarded  as  unwarranted. 

4  The  Alphabet,  London.  1883,  Vol.  II,  pp.  205,  266,  and  The  Acad- 
emy of  Jan.  28,  1882. 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE    31 

have  been  criticized  as  being  affected  by  liis  theory.   The 
following  is  part  of  the  hypothesis : 1 


Numeral 

Hindu 

Bactrian 

Sanskrit 

4 

¥ 

-p     =ch 

chatur,  Lat.  quattuor 

5 

h 

)r  =p 

panclia,  Ok.  irivre 

6 

7 

7 

7   =s 

\  The  s  and  s  a: 

1       occasional! 

sapta  J 

Biihler2  rejects  this  hypothesis,  stating  that  in  four 
cases  (four,  six,  seven,  and  ten)  the  facts  are  absolutely 
against  it. 

While  the  relation  to  ancient  Bactrian  forms  has  been 
generally  doubted,  it  is  agreed  that  most  of  the  numerals 
resemble  Brahnri  letters,  and  we  would  naturally  expect 
them  to  be  initials.3  But,  knowing  the  ancient  pronunci- 
ation of  most  of  the  number  names,4  we  find  this  not 
to  be  the  case.    We  next  fall  back  upon  the  hypothesis 

1  Taylor,  The  Alphabet,  loc.  cit.,  table  on  p.  266. 

2  Biihler,  On  the  Origin  of  the  Indian  Brahma  Alphabet,  Strassburg, 
1898,  footnote,  pp.  52,  53. 

3  Albrecht  Weber,  History  of  Indian  Literature,  English  ed.,  Bos- 
ton, 1878,  p.  256 :  "  The  Indian  figures  from  1-9  are  abbreviated  forms 
of  the  initial  letters  of  the  numerals  themselves  .  .  . :  the  zero,  too, 
has  arisen  out  of  the  first  letter  of  the  word  sunya  (empty)  (it  occurs 
even  in  Pihgala).  It  is  the  decimal  place  value  of  these  figures  which 
gives  them  significance."  C.  Henry,  "  Sur  l'origine  de  quelques  nota- 
tions mathdmatiques,"  Revue  ArcMologique,  June  and  July,  1879,  at- 
tempts to  derive  the  Boethian  forms  from  the  initials  of  Latin  words. 
See  also  J.  Prinsep,  "Examination  of  the  Inscriptions  from  Gimar  in 
Gujerat,  and  Dhauli  in  Cuttach,"  Journal  of  the  Asiatic  Society  of  Ben- 
gal, 1838,  especially  Plate  XX,  p.  348  ;  this  was  the  first  work  on  the 
subject. 

*  Biihler,  Palaeographie,  p.  75,  gives  the  list,  with  the  list  of  letters 
(p.  76)  corresponding  to  the  number  symbols, 


32  TPIE  HINDU- ARABIC  NUMERALS 

that  they  represent  the  order  of  letters  1  in  the  ancient 
alphabet.  From  what  we  know  of  this  order,  however, 
there  seems  also  no  basis  for  this  assumption.  We  have, 
therefore,  to  confess  that  we  are  not  certain  that  the 
numerals  were  alphabetic  at  all,  and  if  they  were  alpha- 
betic we  have  no  evidence  at  present  as  to  the  basis  of 
selection.  The  later  forms  may  possibly  have  been  alpha- 
betical expressions  of  certain  syllables  called  aksaras, 
which  possessed  in  Sanskrit  fixed  numerical  values,2  but 
this  is  equally  uncertain  with  the  rest.  Bay  ley  also 
thought3  that  some  of  the  forms  were  Phoenician,  as 
notably  the  use  of  a  circle  for  twenty,  but  the  resem- 
blance is  in  general  too  remote  to  be  convincing. 

There  is  also  some  slight  possibility  that  Chinese  influ- 
ence is  to  be  seen  in  certain  of  the  early  forms  of  Hindu 
numerals.4 

1  For  a  general  discussion  of  the  connection  between  the  numerals 
and  the  different  kinds  of  alphabets,  see  the  articles  by  U.  Ceretti, 
"Sulla  origine  delle  cifre  numerali  moderne,"  liivista  difisica,  mate- 
matica  e  scienze  naturali,  Pisa  and  Pavia,  1909,  anno  X,  numbers  114, 
118,  119,  and  120,  and  continuation  in  1910. 

2  This  is  one  of  Bidder's  hypotheses.  See  Bayley,  loc.  cit.,  reprint 
p.  4 ;  a  good  bibliography  of  original  sources  is  given  in  this  work,  p.  38. 

3  Loc.  cit.,  reprint,  part  I,  pp.  12,  17.  See  also  Burnell,  loc.  cit., 
p.  64,  and  tables  in  plate  XXIII. 

4  This  was  asserted  by  G.  Hager  (Memoria  sulle  cifre  arabiche, 
Milan,  1813,  also  published  in  Fundgruben  des  Orients,  Vienna,  1811, 
and  in  Bibliotheque  Britannique,  Geneva,  1812).  See  also  the  recent 
article  by  Major  Charles  E.  Woodruff,  "The  Evolution  of  Modern 
Numerals  from  Tally  Marks,"  American  Mathematical  Monthly,  August- 
September,  1909.  Biernatzki,  "  Die  Arithmetik  der  Chinesen,"  Crelle's 
Journal  fur  die  reine  und  angewandte  Mathematik,  Vol.  LII,  1857, 
pp.  59-96,  also  asserts  the  priority  of  the  Chinese  claim  for  a  place 
system  and  the  zero,  but  upon  the  flimsiest  authority.  Ch.  de  Para- 
vey,  Essai  sur  V origine  unique  et  hie'roglyphique  des  chiffres  et  des  Icttres 
detous  les  peuples,  Paris,  1826;  G.  Kleinwachter,  "The  Origin  of  the 
Arabic  Numerals,"  China  Review,  Vol.  XI,  1882-1883,  pp.  379-381, 
Vol.  XII,  ]>]).  28-30;  Biot,  "Note  sur  la  connaissance  que  les  Chinois 
out  eue  lie  la  valeur  de  position  des  chiffres,"  Journal  Asiatiquc,  1839, 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE    33 

More  absurd  is  the  hypothesis  of  a  Greek  origin,  sup- 
posedly supported  by  derivation  of  the  current  symbols 
from  the  first  nine  letters  of  the  Greek  alphabet.1  This 
difficult  feat  is  accomplished  by  twisting  some  of  the 
letters,  cutting  off,  adding  on,  and  effecting  other  changes 
to  make  the  letters  fit  the  theory.  This  peculiar  theory 
was  first  set  up  by  Dasypodius  2  (Conrad  Rauhfuss),  and 
was  later  elaborated  by  Huet.3 

pp.  497-502.  A.  Terrien  de  Lacouperie,  "The  Old  Numerals,  the 
Counting-Rods  and  the  Swan-Pan  in  China,"  Numismatic  Chronicle, 
Vol.  111(3),  pp.  297-340,  and  Crowder  B.  Moseley,  "Numeral  Char- 
acters :  Theory  of  Origin  and  Development,"  American  Antiquarian, 
Vol.  XXII,  pp.  279-284,  both  propose  to  derive  our  numerals  from 
Chinese  characters,  in  much  the  same  way  as  is  done  by  Major  Wood- 
ruff, in  the  article  above  cited. 

1  The  Greeks,  probably  following  the  Semitic  custom,  used  nine 
letters  of  the  alphabet  for  the  numerals  from  1  to  9,  then  nine  others 
for  10  to  90,  and  further  letters  to  represent  100  to  900.  As  the  ordi- 
nary Greek  alphabet  was  insufficient,  containing  only  twenty-four 
letters,  an  alphabet  of  twenty-seven  letters  was  used. 

2  Institutiones  mathematicae,  2  vols.,  Strassburg,  1593-1596,  a  some- 
what rare  work  from  which  the  following  quotation  is  taken  : 

"  Quis  est  harum  Cyphrarum  autor  ? 

"  A  quibus  hae  usitatae  syphrarum  notae  sint  inventae :  hactenus 
incertum  fuit :  meo  tamen  iudicio,  quod  exiguum  esse  fateor :  a  grae- 
cis  librarijs  (quorum  olim  magna  fuit  copia)  literae  Graecorum  quibus 
veteres  Graeci  tamquam  numerorum  notis  sunt  usi :  f  uerunt  corruptae. 
vt  ex  his  licet  videre. 

"  Graecorum  Literae  corruptae. 

-,  "  Sed  qua  ratione  graecorum 

r  f~  cl    t  5  7  /V/->5         literae  ita  fuerunt  corruptae  f 

y  "  Finxerunt    has    corruptas 

/    C  V     J    &    &  <C  V    V         Graecorum  literarum  notas:  vel 

.  abiectione  vt  in  nota  binarij  nu- 

/    2.    3  T"  t)   6>  7  &   /  meri,  vel  additione  vt  in  terna- 

rij,  vel  inuersione  vt  in  septe- 
narij,  numeri  nota,  nostrae  notae,  quibus  hodie  utimur:  ab  his  sola 
differunt  elegantia,  vt  apparet." 

See  also  Bayer,  Ristoria  regni  Graecorum  Bactriani,  St.  Petersburg, 
1738,  pp.  129-130,  quoted  by  Martin,  Recherches  nouvelles,  etc.,  loc.  cit. 

3  P.  D.  Huet,  Demonstrate  evangelica,  Paris,  17G9,  note  to  p.  139  on 
p.  047 :  "Ab  Arabibus  vel  ab  Indis  inventas  esse,  non  vulgus  eruditorum 


34  THE  HINDU-ARABIC  NUMERALS 

A  bizarre  derivation  based  upon  early  Arabic  (c.  1040 
a.d.)  sources  is  given  by  Kircher  in  his  work  1  on  number 
mysticism.  He  quotes  from  Abenragel,2  giving  the  Ara- 
bic and  a  Latin  translation  3  and  stating  that  the  ordinary 
Arabic  forms  are  derived  from  sectors  of  a  circle,  ® . 

Out  of  all  these  conflicting  theories,  and  from  all  the 
resemblances  seen  or  imagined  between  the  numerals  of 
the  West  and  those  of  the  East,  what  conclusions  are  we 
prepared  to  draw  as  the  evidence  now  stands  ?  Probably 
none  that  is  satisfactory.    Indeed,  upon  the  evidence  at 

modo,  sed  doctissimi  quique  ad  hanc  diem  arbitrati  sunt.  Ego  vero 
falsum  id  esse,  merosque  esse  Graecorum  characteres  aio ;  a  librariis 
Graecae  linguae  ignaris  interpolates,  et  diuturna  scribendi  consuetu- 
dine  corruptos.  Nam  primum  i  apex  fuit,  seu  virgula,  nota  fxovaSos.  2, 
est  ipsum  p  extremis  suis  truncatum.  y,  si  in  sinistram  partem  incli- 
naveris  &  cauda  mutilaveris  &  sinistrum  cornu  sinistrorsum  flexeris, 
Set  5.  Res  ipsa  loquitur  4  ipsissimum  esse  A,  cujus  cms  sinistrum 
erigitur  Kara  Kaderov,  &  infra  basim  descendit ;  basis  vero  ipsa  ultra 
crus  producta  eminet.  Vides  quam  5  simile  sit  r£  S\  infimo  tantum 
semicirculo,  qui  sinistrorsum  patebat,  dextrorsum  converse  iiria-rj/xov 
Pav  quod  ita  notabatur  £,  rotundato  ventre,  pede  detracto,  peperit  to  6. 
Ex  Z  basi  sua  mutilato,  ortum  est  rd  7.  Si  H  inflexis  introrsum  api- 
cibus  in  rotundiorem  &  commodiorem  formam  mutaveris,  exurget  to  8. 
At  9  ipsissimum  est  #." 

I.  Weidler,  Spicilegium  observationum  ad  historiam  notarum  nu- 
rneralium,  Wittenberg,  1755,  derives  them  from  the  Hebrew  letters; 
Dom  Augustin  Calmet,  "Recherches  sur  l'origine  des  chiffres  d'arith- 
me^ique,"  Mdmoires  pour  Vhistoire  des  sciences  et  des  beaux  arts,  Tre- 
voux,  1707  (pp.  1020-1035,  with  two  plates),  derives  the  current  symbols 
from  the  Romans,  stating  that  they  are  relics  of  the  ancient  "  Notae 
Tironianae."  These  "  notes"  were  part  of  a  system  of  shorthand  in- 
vented, or  at  least  perfected,  by  Tiro,  a  slave  who  was  freed  by  Cicero. 
L.  A.  Sedillot,  "Sur  l'origine  de  nos  chiffres,"  Atti  delV Accademia 
pontificia  del  nuovi  Lincei,  Vol.  XVIII,  1804-1805,  pp.  310-322,  derives 
the  Arabic  forms  from  the  Roman  numerals. 

1  Athanasius  Kircher,  Arithmologia  sive  De  abditis  Numerorum 
mysterijs  qua  origo,  antiquitas  &  fabrica  Numerorum  exponitur,  Rome, 
1005. 

2  See  Suter,  Die  Malhematiker  und  Astronomen  der  Araber,  p.  100. 

3  "Et  hi  numeri  sunt  numeri  Indiani,  a  Brachmanis  Indiae  Sapi- 
entibus  ex  figura  circuli  secti  inueuti," 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE    35 

hand  we  might  properly  feel  that  everything  points  to 
the  numerals  as  being  substantially  indigenous  to  India. 
And  why  should  this  not  be  the  case  ?  If  the  king 
Srong-tsan-Gampo  (639  a.d.),  the  founder  of  Lhasa,1 
could  have  set  about  to  devise  a  new  alphabet  for  Tibet, 
and  if  the  Siamese,  and  the  Singhalese,  and  the  Burmese, 
and  other  peoples  in  the  East,  could  have  created  alpha- 
bets of  their  own,  why  should  not  the  numerals  also  have 
been  fashioned  by  some  temple  school,  or  some  king,  or 
some  merchant  guild  ?  By  way  of  illustration,  there  are 
shown  in  the  table  on  page  36  certain  systems  of  the 
East,  and  while  a  few  resemblances  are  evident,  it  is 
also  evident  that  the  creators  of  each  system  endeavored 
to  find  original  forms  that  should  not  be  found  in  other 
systems.  This,  then,  would  seem  to  be  a  fair  interpreta- 
tion of  the  evidence.  A  human  mind  cannot  readily 
create  simple  forms  that  are  absolutely  new ;  what  it 
fashions  will  naturally  resemble  what  other  minds  have 
fashioned,  or  what  it  has  known  through  hearsay  or 
through  sight.  A  circle  is  one  of  the  world's  common 
stock  of  figures,  and  that  it  should  mean  twenty  in  Phoe- 
nicia and  in  India  is  hardly  more  surprising  than  that 
it  signified  ten  at  one  time  in  Babylon.2  It  is  therefore 
quite  probable  that  an  extraneous  origin  cannot  be  found 
for  the  very  sufficient  reason  that  none  exists. 

Of  absolute  nonsense  about  the  origin  of  the  sym- 
bols which  we  use  much  has  been  written.    Conjectures, 

1  V.  A.  Smith,  The  Early  History  of  India,  Oxford,  2d  ed.,  1908, 
p.  333. 

2  C.  J.  Ball,  "An  Inscribed  Limestone  Tablet  from  Sippara,"  Pro- 
ceedings of  the  Society  of  Biblical  Archaeology,  Vol.  XX,  p.  25  (Lon- 
don, 1808).  Terrien  de  Laconperie  states  that  the  Chinese  used  the 
circle  for  10  before  the  beginning  of  the  Christian  era.  [Catalogue  of 
Chinese  Coins,  London,  1892,  p.  xl.] 


Siam 
2  Burma 
»  Malabar 

4  Tibet 


36  THE  HINDU-ARABIC  NUMERALS 

however,  without  any  historical  evidence  for  support, 
have  no  place  in  a  serious  discussion  of  the  gradual  evo- 
lution of  the  present  numeral  forms.1 


Tablk  of  Certain  Eastern  Systems 

0  1  2  345G789  10 

o   s*  re  fZ./ft>£)yj[j&rx  jjj 

i  z  ?&  vara  ?  x  rr 

Ceylon  ^    ^O    6h^  ^  Qw  O  ^^©V      ity 

•Malayalam  <°       d_    °X    &     @  '   "fl     ^ »/\_J     fy^   jjj 

1  For  a  purely  fanciful  derivation  from  the  corresponding  number 
of  strokes,  see  W.  W.  R.  Ball,  A  Short  Account  of  the  History  of  Mathe- 
matics, 1st  ed.,  London,  1888,  p.  147  ;  similarly  J.  B.  Reveillaud,  Essai 
sur  lea  chiffres  arabes,  Paris,  1883;  P.  Voizot,  "Les  chiffres  arabes  et 
leur  origine,"  La  Nature,  1899,  p.  222  ;  G.  Dumesnil,  "De  la  forme  des 
chiffres  usuels,"  Annates  de  Vuniversite"  de  Grenoble,  1907,  Vol.  XIX, 
pp.  657-674,  also  a  note  in  Revue  Archeologique,  1890,  Vol.  XVI  (3), 
pp.  342-348;  one  of  the  earliest  references  to  a  possible  derivation 
from  points  is  in  a  work  by  Bettino  entitled  Apiaria  universae  philo- 
sophiae  mathematicae  in  quibus  paradoxa  et  noua  machinamenta  ad  usus 
eximios  traducta,  et  facillimis  demonstrationibus  confirmaia,  Bologna 
1545,  Vol.  II,  Apiarium  XI,  p.  5. 

2  Alphabetum  Barmanum,  Romae,  mdcclxxvi,  p.  50.  The  1  is  evi- 
dently Sanskrit,  and  the  4,  7,  and  possibly  9  are  from  India. 

3  Alphabetum  Grandonico-Malabaricum,  Romae,  sidcclxxii,  p.  90. 
The  zero  is  not  used,  but  the  symbols  for  10,  100,  and  so  on,  are  joined 
to  the  units  to  make  the  higher  numbers. 

4  Alphabetum  Tangutanum,  Romae,  mdcclxxiii,  p.  107.  In  a  Ti- 
betan MS.  in  the  library  of  Professor  Smith,  probably  of  the  eigh- 
teenth century,  substantially  these  forms  are  given. 

5  Bayley,  loc.  cit.,  plate  II.  Similar  forms  to  these  here  shown,  and 
numerous  other  forms  found  in  India,  as  well  as  those  of  other  oriental 
countries,  are  given  by  A.  P.  Pilian,  Expose  des  signes  de  nume'ratim, 
usitds  chcz  les  peuplcs  oricntaux  anciens  ct  modernes,  Paris,  1860. 


EARLY  HINDU  FORMS  WITH  NO  PLACE  VALUE     37 

We  may  summarize  this  chapter  by  saying  that  no  one 
knows  what  suggested  certain  of  the  early  numeral  forms 
used  in  India.  The  origin  of  some  is  evident,  but  the 
origin  of  others  will  probably  never  be  known.  There  is 
no  reason  why  they  should  not  have  been  invented  by 
some  priest  or  teacher  or  guild,  by  the  order  of  some 
king,  or  as  part  of  the  mysticism  of  some  temple.  What- 
ever the  origin,  they  were  no  better  than  scores  of  other 
ancient  systems  and  no  better  than  the  present  Chinese 
system  when  written  without  the  zero,  and  there  would 
never  have  been  any  chance  of  their  triumphal  progress 
westward  liad  it  nol  been  for  1  his  relatively^B^j symbol. 
There  could  hardly  be  demanded  a  strongeflj  Pof  of  the 
Hindu  origin  of  the  character  for  zero  than  this,  and  to 
it  further  reference  will  be  made  in  Chapter  IVo 


CHAPTER  III 
LATER  HINDU  FORMS,  WITH  A  PLACE  VALUE 

Before  speaking  of  the  perfected  Hindu  numerals  with 
the  zero  and  the  place  value,  it  is  necessary  to  consider 
the  third  system  mentioned  on  page  19,  —  the  word  and 
letter  forms.  The  use  of  words  with  place  value  began 
at  least  as  early  as  the  6th  century  of  the  Christian  era. 
In  many  of  the  manuals  of  astronomy  and  mathematics, 
and  often  in  other  works  in  mentioning  dates,  numbers 
are  represented  by  the  names  of  certain  objects  or  ideas. 
For  example,  zero  is  represented  by  "  the  void  "  (sunya), 
or  "  heaven-space "  (ambara  dkdki)  ;  one  by  "  stick  " 
(rwpa),  "  moon  "  (indii  sasiri),  "  earth  "  (bhu),  "  begin- 
ning "  (adi),  "  Brahma,"  or,  in  general,  by  anything 
markedly  unique ;  two  by  "  the  twins  "  Qyamd),  "  hands  " 
(hard),  "  eyes  "  (nayand),  etc. ;  four  by  "  oceans,"  five 
by  "senses"  (yimya)  or  "arrows"  (the  five  arrows  of 
Kamadeva)  ;  six  by  "seasons"  or  "flavors";  seven  by 
"mountain  "  (ago),  and  so  on.1  These  names,  accommo- 
dating themselves  to  the  verse  in  which  scientific  works 
were  written,  had  the  additional  advantage  of  not  admit- 
ting, as  did  the  figures,  easy  alteration,  since  any  change 
would  tend  to  disturb  the  meter. 

1  Biihler,  loc.  cit.,  p.  80;  J.  F.  Fleet,  Corpus  inscriptioniun  Tndica- 
rum,  Vol.  Ill,  Calcutta,  1888.  Lists  of  such  words  are  given  also  by 
Al-Birunl  in  his  work  India;  by  Burnell,  loc.  cit.;  by  E.  Jacquet, 
"Mode  d'expression  symboliquc  des  nombres  employe"  par  les  Indiens, 
lesTibelains  et  les  Javanais,"  Journal  Asiatique,  Vol.  XVI,  Paris,  1835. 

38 


LATER  HINDU  FORMS  WITH  A  PLACE  VALUE     39 

As  an  example  of  this  system,  the  date  "  Saka  Samvat, 
867"  (a.d.  945  or  940),  is  given  by  " giri-rasa-vasu" 
meaning  "  the  mountains  "  (seven),  "  the  flavors  "  (six), 
and  the  gods  "  Vasu  "  of  which  there  were  eight.  In  read- 
ing the  date  these  are  read  from  right  to  left.1  The 
period  of  invention  of  this  system  is  uncertain.  The  first 
trace  seems  to  be  in  the  Srautasiitra  of  Katyayana  and 
Latyayana.2  It  was  certainly  known  to  Varaha-Mihira 
(d.  58  7),3  for  he  used  it  in  the  Brhat-Samhitd.4  It  has 
also  been  asserted5  that  Aryabhata  (c.  500  A.D.)  was 
familiar  with  this  system,  but  there  is  nothing  to  prove 
the  statement.6  The  earliest  epigraphical  examples  of 
the  system  are  found  in  the  Bayang  (Cambodia)  inscrip- 
tions of  604  and  624  a.d.7 

Mention  should  also  be  made,  in  this  connection,  of  a 
curious  system  of  alphabetic  numerals  that  sprang  up  in 
southern  India.  In  this  we  have  the  numerals  repre- 
sented by  the  letters  as  given  in  the  following  table : 


1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

k . 

kh 

gh 

ri 

c 

ch 

J 

jh 

n 

t 

th 

(1 

dh 

n 

t 

th 

d 

dh 

n 

p 

ph 

b 

bh 

m 

y 

r 

1 

V 

s 

s 

s 

h 

1 

1  This  date  is  given  by  Fleet,  loc.  cit.,  Vol.  Ill,  p.  73,  as  the  earliest 
epigraphical  instance  of  this  usage  in  India  proper. 

2  Weber,  Indische  Studien,  Vol.  VIII,  p.  166 seq. 

3  Journal  of  the  Royal  Asiatic  Society,  Vol.  I  (n.s.),  p.  407. 
*  VIII,  20,  21. 

5  Th.  II.  Martin,  Les  signes  numeraux  .  .  .,  Rome,  1864;  Lassen, 
Indische  Alterthumskunde,  Vol.  II,  2d  ed.,  Leipzig  and  London,  1874, 
p.  1153. 

6  But  see  Burnell,  loc.  cit.,  and  Thibaut,  Astronomie,  Astrologie  und 
Mathematik,  p.  71. 

7  A.  Barth,  "  Inscriptions  Sanscrites  du  Cambodge,"  in  the  Notices  et 
extraits  des  Mss.  de  la  Bihliotheque  nationale,  Vol.  XXVII,  Part  I,  pp.  1- 
180, 1885;  see  also  numerous  articles  in  Journal  Asiatique,  by  Aymonier. 


40  THE  HINDU-ARABIC  NUMERALS 

By  this  plan  a  numeral  might  be  represented  by  any 
one  of  several  letters,  as  shown  in  the  preceding  table, 
and  thus  it  could  the  more  easily  be  formed  into  a  word 
for  mnemonic  purposes.    For  example,  the  word 

2       3  1  5       6      5        1 

kha  gont  yan  me  sa  ma  pa 

has  the  value  1,565,132,  reading  from  right  to  left.1  This, 
the  oldest  specimen  (1184  a.d.)  known  of  this  notation, 
is  given  in  a  commentary  on  the  Rigveda,  representing 
the  number  of  days  that  had  elapsed  from  the  beginning 
of  the  Kaliyuga.  Burnell  2  states  that  this  system  is 
even  yet  in  use  for  remembering  rules  to  calculate  horo- 
scopes, and  for  astronomical  tables. 

A  second  system  of  this  kind  is  still  used  in  the 
pagination  of  manuscripts  in  Ceylon,  Siam,  and  Burma, 
having  also  had  its  rise  in  southern  India.  In  this  the 
thirty-four  consonants  when  followed  by  a  (as  ha  .  .  .  la) 
designate  the  numbers  1-34 ;  by  «  (as  ltd  .  .  .  /a),  those 
from  35  to  68 ;  by  i  (Jci  .  .  .  li),  those  from  69  to  102, 
inclusive  ;  and  so  on.3 

As  already  stated,  however,  the  Hindu  system  as  thus 
far  described  was  no  improvement  upon  many  others  of 
the  ancients,  such  as  those  tlsed  by  the  Greeks  and  the 
Hebrews.  Having  no  zero,  it  was  impracticable  to  desig- 
nate the  tens,  hundreds,  and  other  units  of  higher  order 
by  the  same  symbols  used  for  the  units  from  one  to  nine. 
In  other  words,  there  was  no  possibility  of  place  value 
without  some  further  improvement.     So  the  Nana  Ghat 

1  Biihler,  loc.  cit.,  p.  82. 

2  Loc.  cit.,  p.  70. 

3  Biihler,  loc.  cit.,  p.  S3.  The  Hindu  astrologers  still  use  an  alpha- 
betical system  of  numerals.    [Burnell,  loc.  cit.,  p.  79.] 


LATER  HINDU  FORMS  WITH  A  PLACE  VALUE     41 

symbols  required  the  writing  of  "  thousand  seven  twenty- 
four  "  about  like  T  7,  tw,  4  in  modern  symbols,  instead 
of  7024,  in  which  the  seven  of  the  thousands,  the  two 
of  the  tens  (concealed  in  the  word  twenty,  being  origi- 
nally "twain  of  tens,"  the  -ty  signifying  ten),  and  the 
four  of  the  units  are  given  as  spoken  and  the  order  of 
the  unit  (tens,  hundreds,  etc.)  is  given  by  the  place.  To 
complete  the  system  only  the  zero  was  needed ;  but  it 
was  probably  eighty  centuries  after  the  Nana  Ghat  inscrip- 
tions were  cut,  before  this  important  symbol  appeared ; 
and  not  until  a  considerably  later  period  did  it  become 
well  known.  Who  it  was  to  whom  the  invention  is  due, 
or  where  he  lived,  or  even  in  what  century,  will  probably 
always  remain  a  mystery.1  It  is  possible  that  one  of  the 
forms  of  ancient  abacus  suggested  to  some  Hindu  astron- 
omer or  mathematician  the  use  of  a  symbol  to  stand  for 
the  vacant  line  when  the  counters  were  removed.  It  is 
well  established  that  in  different  parts  of  India  the  names 
of  the  higher  powers  took  different  forms,  even  the  order 
being  interchanged.2  Nevertheless,  as  the  significance  of 
the  name  of  the  unit  was  given  by  the  order  in  reading, 
these  variations  did  not  lead  to  error.  Indeed  the  varia- 
tion itself  may  have  necessitated  the  introduction  of  a 
word  to  signify  a  vacant  place  or  lacking  unit,  with  the 
ultimate  introduction  of  a  zero  symbol  for  this  word. 

To  enable  us  to  appreciate  the  force  of  this  argument 
a  large  number,  8,443,682,155,  may  be  considered  as  the 
Hindus  wrote  and  read  it,  and  then,  by  way  of  contrast, 
as  the  Greeks  and  Arabs  would  have  read  it. 

1  Well  could  Ramus  say,  "Quicunq;  autein  fuerit  inventor  decern 
notarum  lauclem  magnam  meruit." 

2  Al-Blrunl  gives  lists. 


42  THE  HINDU-ARABIC  NUMERALS 

Modern  American  reading,  8  billion,  443  million,  682 
thousand,  155. 

Hindu,  8  paclmas,  4  vyarbudas,  4  kotis,  3  prayutas, 
6  laksas,  8  ayutas,  2  sahasra,  1  sata,  5  dasan,  5. 

Arabic  and  early  German,  eight  thousand  thousand 
thousand  and  four  hundred  thousand  thousand  and  forty- 
three  thousand  thousand,  and  six  hundred  thousand  and 
eighty-two  thousand  and  one  hundred  fifty -five  (or  five 
and  fifty). 

G-reek,  eighty-four  myriads  of  myriads  and  four  thou- 
sand three  hundred  sixty-eight  myriads  and  two  thou- 
sand and  one  hundred  fifty-five. 

As  Woepcke  1  pointed  out,  the  reading  of  numbers  of 
this  kind  shows  that  the  notation  adopted  by  the  Hindus 
tended  to  bring  out  the  place  idea.  No  other  language 
than  the  Sanskrit  has  made  such  consistent  application, 
in  numeration,  of  the  decimal  system  of  numbers.  The 
introduction  of  myriads  as  in  the  Greek,  and  thousands 
as  in  Arabic  and  in  modern  numeration,  is  really  a  step 
away  from  a  decimal  scheme.  So  in  the  numbers  below 
one  hundred,  in  English,  eleven  and  twelve  are  out  of 
harmony  with  the  rest  of  the  -teens,  while  the  naming  of 
all  the  numbers  between  ten  and  twenty  is  not  analogous 
to  the  naming  of  the  numbers  above  twenty.  To  conform 
to  our  written  system  we  should  have  ten-one,  ten-two, 
ten-three,  and  so  on,  as  we  have  twenty-one,  twenty-two, 
and  the  like.  The  Sanskrit  is  consistent,  the  units,  how- 
ever, preceding  the  tens  and  hundreds.  Nor  did  any. 
other  ancient  people  carry  the  numeration  as  far  as  did 
the  Hindus.2 

1  Propagation,  loc.  cit.,  p.  443. 

-  See  the  quotation  from  The  Light  of  Asia  in  Chapter  II,  p.  1G. 


LATER  HINDU  FORMS  WITH  A  PLACE  VALUE     43 

When  the  anJcapalli,1  the  decimal-place  system  of  writ- 
ing numbers,  was  perfected,  the  tenth  symbol  was  called  / 
the  sunyahhulu,  generally  shortened  to  rnnya  (the  void). 
Broekhaus  2  has  well  said  that  if  there  was  any  invention 
for  which  the  Hindus,  by  all  their  philosophy  and  reli- 
gion, were  well  fitted,  it  was  the  invention  of  a  symbol 
for  zero.  This  making  of  nothingness  the  crux  of  a  tre- 
mendous achievement  was  a  step  in  complete  harmony 
with  the  genius  of  the  Hindu. 

It  is  generally  thought  that  this  mnya  as  a  symbol 
was  not  used  before  about  j500_A-p.,  although  some  writ- 
ers have  placed  it  earlier.3  Since  Aryabhata  gives  our 
common  method  of  extracting  roots,  it  would  seem  that 
he  may  have  known  a  decimal  notation,4  although  he 
did  not  use  the  characters  from  which  our  numerals 
are  derived.5     Moreover,   he   frequently  speaks   of  the 

1  The  nine  ciphers  were  called  anka. 

2  "  Zur  Geschichte  des  indischen  Ziffernsystems,"  Zeitschrift  fur  die 
Kunde  des  Morgenlandes,  Vol.  IV,  1842,  pp.  74-83. 

3  It  is  found  in  the  Bakhsall  MS.  of  an  elementary  arithmetic 
which  Hoernle  placed,  at  first,  about  the  beginning  of  our  era,  but  the 
date  is  much  in  question.  G.  Thibaut,  loc.  cit.,  places  it  between  700 
and  900  A.D. ;  Cantor  places  the  body  of  the  work  about  the  third  or 
fourth  century  a.i>.,  Geschichte  der  Mathematik,  Vol.  I  (3),  p.  598. 

4  For  the  opposite  side  of  the  case  see  G.  R.  Kaye,  "Notes  on  Indian 
Mathematics,  No.  2. —  Aryabhata,"  Joum.  and  Proc.  of  the  Asiatic  Soc. 
of  Bengal,  Vol.  IV,  1908,  pp.  111-141. 

5  He  used  one  of  the  alphabetic  systems  explained  above.  This  ran 
up  to  1018  and  was  not  difficult,  beginning  as  follows : 

"3>       ^T/      "^     ^>      *£ 

1  102         104        toe         108;  etc., 

the  same  letter  (ka)  appearing  in  the  successive  consonant  forms,  ka, 
kha,  ga,  gha,  etc.  See  C.  I.  Gerhard t,  Uber  die  Entstehung  und  Aus- 
breitung  des  dekadischen  Zahlensystcms,  Programm,  p.  17,  Salzwedel, 
1853,  and  Etudes  historiques  sur  V  arithmetique  de  position,  Programm, 
p.  24,  Berlin,  1856;  E.  Jacquet,  Mode  d' expression symboliquedes nombres, 


44  THE  HINDU-ARABIC  NUMERALS 

void.1  If  he  refers  to  a  symbol  this  would  put  the  zero 
as  far  back  as  500  a.d.,  but  of  course  he  may  have  re- 
ferred merely  to  the  concept  of  nothingness. 

A  little  later,  but  also  in  the  sixth  century,  Varaha- 
Mihira2  wrote  a  work  entitled  Brhat  Samhitd3  in  which 
he  frequently  uses  iunya  in  speaking  of  numerals,  so 
that  it  has  been  thought  that  he  was  referring  to  a  defi- 
nite symbol.  This,  of  course,  would  add  to  the  proba- 
bility that  Aryabhata  was  doing  the  same. 

It  should  also  be  mentioned  as  a  matter  of  interest,  and 
somewhat  related  to  the  cprestion  at  issue,  that  Varaha- 
Mihira  used  the  word-system  with  place  value4  as  ex- 
plained above. 

The  first  kind  of  alphabetic  numerals  and  also  the 
word-system  (in  both  of  which  the  place  value  is  used) 
are  plays  upon,  or  variations  of,  position  arithmetic,  which 
would  be  most  likely  to  occur  in  the  country  of  its  origin.5 

At  the  opening  of  the  next  century  (c.  620  a.d.)  Bana  6 
wrote  of  Subandhus's  Vdsavadattd  as  a  celebrated  work, 

loc.  cit.,  p.  97  ;  L.  Rodet/'Sur  la  veritable  signification  de  la  notation 
num6rique  invents  par  Aryabhata,1 '  Journal  Asiatique,  Vol.  XVI  (7), 
pp.  440-485.  On  the  two  Aryabhatas  see  Kaye,  Bibl.  Math.,  Vol.  X  (3), 
p.  289. 

1  Using  kha,  a  synonym  of  sunya.  [Bayley,  loc.  cit.,  p.  22,  and  L. 
Bodet,  Journal  Asiatique,  Vol.  XVI  (7),  p.  443.] 

2  Varaha-Mihira,  Pancasiddhantika,  translated  by  G.  Thibant  and 
M.  S.  Dvivedi,  Benares,  1889;  see  Buhler,  loc.  cit.,  p.  78;  Bayley, 
loc.  cit.,  p.  23. 

3  Brhat  Sarnhita,  translated  by  Kern,  Journal  of  the  Royal  Asiatic 
Society,  1870-i875. 

4  It  is  stated  by  Buhler  in  a  personal  letter  to  P.ayley  (Inc.  cit.,  p.  0.r>) 
that  there  are  hundreds  of  instances  of  tins  usage  in  the  Brhat  Sarn- 
hita. The  system  was  also  used  in  the  Pancasiddhantika  as  early  as 
505  a.d.  [Buhler,  Palaeographie,  p.  80,  and  Fleet,  Journal  of  tin  Royal 
Asiatic  Society,  1910,  p.  819.] 

6  Cantor,  Geschichte  der  Mathematik,  Vol.  I  (3),  p.  008. 
6  Biihler,  loc.  cit.,  p.  78. 


LATER  HINDU  FORMS  WITH  A  PLACE  VALUE     45 

and  mentioned  that  the  stars  dotting  the  sky  are  here 
compared  with  zeros,  these  being  points  as  in  the  mod- 
ern Arabic  system.  On  the  other  hand,  a  strong  argu- 
ment against  any  Hindu  knowledge  of  the  symbol  zero 
at  this  time  is  the  fact  that  about  700  A.D.  the  Arabs 
overran  the  province  of  Sind  and  thus  had  an  opportu- 
nity of  knowing  the  common  methods  used  there  for 
writing  numbers.  And  yet,  when  they  received  the  com- 
plete system  in  776  they  looked  upon  it  as  something 
new.1  Such  evidence  is  not  conclusive,  but  it  tends  to 
show  that  the  complete  system  was  probably  not  in  com- 
mon use  in  India  at  the  beginning  of  the  eighth  century. 
On  the  other  hand,  we  must  bear  in  mind  the  fact  that 
a  traveler  in  Germany  in  the  year  1700  would  probably 
have  heard  or  seen  nothing  of  decimal  fractions,  although 
these  were  perfected  a  century  before  that  date.  The 
elite  of  the  mathematicians  may  have  known  the  zero 
even  in  Aryabhata's  time,  while  the  merchants  and  the 
common  people  may  not  have  grasped  the  significance  of 
the  novelty  until  a  long  time  after.  On  the  whole,  the 
evidence  seems  to  point  to  the  west  coast  of  India  as  the 
region  where  the  complete  system  was  first  seen.2  As 
mentioned  above,  traces  of  the  numeral  words  with  place 
value,  which  do  not,  however,  absolutely  require  a  deci- 
mal place-system  of  symbols,  are  found  very  early  in 
Cambodia,  as  well  as  in  India. 

Concerning  the  earliest  epigraphical  instances  of  the  use 
of  the  nine  symbols,  plus  the  zero,  with  place  value,  there 

i  Bayley,  p.  38. 

2  Noviomagus,  in  his  De  numeris  libri  duo,  Paris,  1539,  confesses  his 
ignorance  as  to  the  origin  of  the  zero,  but  says :  "  D.  Henricus  Grauius, 
vir  Graece  &  Hebraic^  exime  doctus,  Hebraicam  originem  ostendit," 
adding  that  Valla  "Indis  Orientalibus  gentibus  inventionem  tribuit." 


46  TUP:   HINDU-ARABIC  NUMERALS 

is  some  question.  Colebrooke1  in  1807  warned  against 
the  possibility  of  forgery  in  many  of  the  ancient  copper- 
plate land  grants.  On  this  account  Fleet,  in  the  Indian 
Antiquary?  discusses  at  length  this  phase  of  the  work  of 
the  epigraphists  in  India,  holding  that  many  of  these 
forgeries  were  made  about  the  end  of  the  eleventh  cen- 
tury.  Colebrooke3  takes  a  more  rational  view  of  these 
forgeries  than  does  Kaye,  who  seems  to  hold  that  they 
tend  to  invalidate  the  whole  Indian  hypothesis.  "But 
even  where  that  may  lie  suspected,  the  historical  uses  of 
a  monument  fabricated  so  much  nearer  to  the  times  to 
which  it  assumes  to  belong,  will  not  be  entirely  super- 
seded. The  necessity  of  rendering  the  forged  grant  credi- 
ble would  compel  a  fabricator  to  adhere  to  history,  and 
conform  to  established  notions  :  and  the  tradition,  which 
prevailed  in  his  time,  and  by  which  he  must  be  guided, 
would  probably  be  so  much  nearer  to  the  truth,  as  it 
was  less  remote  from  the  period  which  it  concerned."  4 
Biihler5  gives  the  copper-plate  Gurjara  inscription  of 
Cedi-samvat  34(3  (595  a.d.)  as  the  oldest  epigraphical 
use  of  the  numerals  6  "  in  which  the  symbols  correspond 
to  the  alphabet  numerals  of  the  period  and  the  place." 
Vincent  A.  Smith7  quotes  a  stone  inscription  of  815  A.D., 
dated  Samvat  872.  So  F.  Kielhorn  in  the  EpigrapHa 
Iit<ttcas  gives  a  Pathari  pillar  inscription  of  Parabala, 
dated  Vikrama-samvat  917,  which  corresponds  to  861  A.D., 

i  See  Esmyz,  Vol.  II,  pp.  287  ami  288. 

2  Vol.  XXX,  p.  205  seqq.  3  Loc.  cit.,  p.  284  seqq. 

4  Colebrooke,  loc.  cit.,  p.  288.  5  Loc.  cit.,  p.  78. 

6  Hereafter,  unless  expressly  stated  to  the  contrary,  we  shall  use 
the  word  "numerals"  to  mean  numerals  with  place  value. 

7  "The  Gurjaras  of  R&jputana  and  Kanauj,"  inJournal  of  the  Royal 
Asiatic  Society,  January  and  April,  190'J. 

»  Vol.  IX,  1908,  p.  248. 


LATER  HINDU  FORMS  WITH  A  PLACE  VALUE     47 

and  refers  also  to  another  copper-plate  inscription  dated 
Vikrama-samvat  813  (756  a.d.).  The  inscription  quoted 
by  V.  A.  Smith  above  is  that  given  by  D.  R.  Bhan- 
darkar,1  and  another  is  given  by  the  same  writer  as  of 
date  Saka-samvat  715  (798  a.d.),  being  incised  on  a 
pilaster.  Kielhorn2  also  gives  two  copper-plate  inscrip- 
tions of  the  time  of  Mahendrapala  of  Kanauj,  Valhabl- 
samvat  574  (893  A.d.)  and  Vikrama-samvat  956  (899 
a.d.).  That  there  should  be  any  inscriptions  of  date  as 
early  even  as  750  A.D.,  would  tend  to  show  that  the  sys- 
tem was  at  least  a  century  older.  As  will  be  shown  in 
the  further  development,  it  was  more  than  two  centu- 
ries after  the  introduction  of  the  numerals  into  Europe 
that  they  appeared  there  upon  coins  and  inscriptions. 
While  Thibaut 3  does  not  consider  it  necessary  to  quote 
any  specific  instances  of  the  use  of  the  numerals,  he 
states  that  traces  are  found  from  590  a.d.  on.  "  That 
the  system  now  in  use  by  all  civilized  nations  is  of  Hindu 
origin  cannot  be  doubted ;  no  other  nation  has  any  claim 
upon  its  discovery,  especially  since  the  references  to  the 
origin  of  the  system  which  are  found  in  the  nations  of 
western  Asia  point  unanimously  towards  India."  4 

The  testimony  and  opinions  of  men  like  Biihler,  Kiel- 
horn,  V.  A.  Smith,  Bhandarkar,  and  Thibaut  are  entitled 
to  the  most  serious  consideration.  As  authorities  on 
ancient  Indian  epigraphy  no  others  rank  higher.  Their 
work  is  accepted  by  Indian  scholars  the  world  over,  and 
their  united  judgment  as  to  the  rise  of  the  system  with 
a  place  value  —  that  it  took  place  in  India  as  early  as  the 

1  Epigraphia  Indica,  Vol.  IX,  pp.  193  and  198. 

2  Epigraphia  Indica,  Vol.  IX,  p.  1. 

3  Loc.  cit.,  p.  71.  4  Thibaut,  p.  71. 


48  THE  HINDU-ARABIC  NUMERALS 

sixth  century  a.d.  —  must  stand  unless  new  evidence  of 
great  weight  can  be  submitted  to  the  contrary. 

Many  early  writers  remarked  upon  the  diversity  of 
Indian  numeral  forms.  Al-Blruni  was  probably  the  first ; 
noteworthy  is  also  Johannes  Hispalensis,1  who  gives  the 
variant  forms  for  seven  and  four.  We  insert  on  p.  49  a 
table  of  numerals  used  with  place  value.  While  the  chief 
authority  for  this  is  Biihler,2  several  specimens  are  given 
which  are  not  found  in  his  work  and  which  are  of  unusual 
interest. 

The  Sarada  forms  given  in  the  table  use  the  circle  as  a 
symbol  for  1  and  the  dot  for  zero.  They  are  taken  from 
the  paging  and  text  of  The  Kashmirian  Atharva-Veda,3 
of  which  the  manuscript  used  is  certainly  four  hundred 
years  old.  Similar  forms  are  found  in  a  manuscript  be- 
longing to  the  University  of  Tubingen.  Two  other  series 
presented  are  from  Tibetan  books  in  the  library  of  one 
of  the  authors. 

For  purposes  of  comparison  the  modern  Sanskrit  and 
Arabic  numeral  forms  arc  added. 


Sanskrit, 


Arabic, 


\rrioi\M. 


1  "  Est  autem  in  aliquibus  figurarum  istarum  apud  multos  diuersi- 
tas.  Quidam  enim  septimam  banc  figuram  representant,"  etc.  [Bon- 
compagni,  Trattati,  p.  28.]  Enestrom  has  shown  that  very  likely  this 
work  is  incorrectly  attributed  to  Johannes  Hispalensis.  [BiUliotheca 
Mathematical,,  Vol.  IX  (3),  p.  2.] 

2  Indische  Palaeographie,  Tafel  IX. 

3  Edited  by  Bloomfield  and  Garbe,  Baltimore,  1901,  containing 
photographic  reproductions  of  the  manuscript. 


LATER  HINDU  FORMS  WITH  A  PLACE  VALUE      49 
Numerals  used  with  Place  Value 

12345G7  890 

"  J-    <2-  3   4  S  S   7    -Y  ?   ° 

k  J    Z  1  f<  9  6  o 

O  3  ?   **   ^5^   • 

a  Bakhsali  MS.  See  page  43 ;  Hoernle,  R.,  The  Indian  Antiquary, 
Vol.  XVII,  pp.  33-48,  1  plate  ;  Hoernle,  Verhandlungen  des  VII.  Inter- 
nationalen  Orientalisten-Congr  esses,  Arische  Section,  Vienna,  1888,  "On 
the  Bakshali  Manuscript,"  pp.  127-147,  3  plates;  Biihler,  loc.  cit. 

b  3,4,6,  from  H.  H.  Dhruva,  "Three  Land-Grants  from  Sankheda," 
Epiyraphia  Indica,  Vol.  II,  pp.  li)-24  with  plates ;  date  505  a.i>.    7, 1, 5, 


50  THE  HINDU-ARABIC  NUMERALS 

from  Bhandarkar,  "  Daulatabad  Plates,"  Epigraphia  Indica,  Vol.  IX, 
part  V  ;  date  c.  798  a.d. 

c  8,  7,  2,  from  "Buckhala  Inscription  of  Nagabhatta,"  Bhandarkar, 
Epigraphia  Indica,  Vol.  IX,  part V  ;  date  815  a.d.  5  from  "The  Morbi 
('upper-Plate,"  Bhandarkar,  The  Indian  Antiquary,  Vol.  II,  pp.  257- 
258,  with  plate;  date  804  a.d.    See  Biihler,  loc.  cit. 

<i  8  from  the  above  Morbi  Copper-Plate.  4,  5,  7,  9,  and  0,  from  "Asni 
Inscription  of  Mahipala,"  The  Indian  Antiquary,  Vol.  XVI,  pp.  174- 
175;  inscription  is  on  red  sandstone,  date  917  a.d.    See  Biihler. 

e  8,  9,  4,  from  "  Rashtrakuta  Grant  of  Amoghavarsha,"  J.  F.  Fleet, 
The  Indian  Antiquary,  Vol.  XII,  pp.  263-272 ;  copper-plate  grant  (if 
date  c.  972  a.d.  See  Biihler.  7,  3,  5,  from  "Torkkede  Copper-Plate 
Grant  of  the  Time  of  Govindaraja  of  Gujerat,"  Fleet,  Epigraphia  In- 
dica, Vol.  Ill,  pp.  53-58.    See  Biihler. 

f  From  "A  Copper-Plate  Grant  of  King  Tritochanapala  Chahlukya 
of  Latadesa,"  H.  H.  Dhruva,  Indian  Antiquary,  Vol.  XII,  pp.  196- 
205;  date  1050  a.d.    See  Biihler. 

e  Burned,  A.  C,  South  Indian  Palaeography,  plate  XXIII,  Telugu- 
Canarese  numerals  of  the  eleventh  century.    See  Biihler. 

h  and  i  From  a  manuscript  of  the  second  half  of  the  thirteenth 
century,  reproduced  in  "  Delia  vita  e  delle  opere  di  Leonardo  Pisano," 
Baldassare  Boncompagni,  Rome,  1852,  in  Atti  deW Accademia  Pontificia 
dei  nuovi  Lincei,  anno  V. 

i  and  k  From  a  fourteenth-century  manuscript,  as  reproduced  in 
Delia  vita  etc.,  Boncompagni,  loc.  cit. 

1  From  a  Tibetan  MS.  in  the  library  of  D.  E.  Smith. 

m  From  a  Tibetan  block-book  in  the  library  of  D.  E.  Smith. 

n  Sarada  numerals  from  The  Kashmirian  Atharva-Veda,  reproduced 
by  chromophotography  from  the  manuscript  in  .the  University  Library 
at  Tubingen,  Bloomfield  and  Garbe,  Baltimore,  1901.  'Somewhat  sim- 
ilar forms  are  given  under  "Numeration  Cachemirienne,"  by  Pihan, 
Expose"  etc.,  p.  84. 


CHAPTER  IV 

THE   SYMBOL   ZERO 

What  has  been  said  of  the  improved  Hindu  system 
with  a  place  value  does  not  touch  directly  the  origin  of 
a  symbol  for  zero,  although  it  assumes  that  such  a  sym- 
bol exists.  The  importance  of  such  a  sign,  the  fact  that  it 
is  a  prerequisite  to  a  place-value  system,  and  the  further 
fact  that  without  it  the  Hindu-Arabic  numerals  would 
never  have  dominated  the  computation  system  of  the 
western  world,  make  it  proper  to  devote  a  chapter  to  its 
origin  and  history. 

It  was  some  centuries  after  the  primitive  Brahmi  and 
Kharosthi  numerals  had  made  their  appearance  in  India 
that  the  zero  first  appeared  there,  although  such  a  char- 
acter was  used  by  the  Babylonians 1  in  the  centuries 
immediately  preceding  the  Christian  era.  The  symbol  is 
^  or  ^,  and  apparently  it  was  not  used  in  calculation. 
Nor  does  it  always  occur  when  units  of  any  order  are 
lacking;  thus  180  is  written  YYY  with  the  meaning 
three  sixties  and  no  units,  since  181  immediately  follow- 
ing is  Y  y  Y     Y  j  three  sixties  and  one  unit.2    The  main 

1  Franz  X.  Kugler,  Die  Babylonische  Mondreehnung,  Freiburg  i.  Br., 
1000,  in  the  numerous  plates  at  the  end  of  the  book ;  practically  all 
of  these  contain  the  symbol  to  which  reference  is  made.  Cantor, 
Geschichte,  Vol.  I,  p.  31. 

2  F.  X.  Kugler,  Sternkunde  und  Sterndienst  in  Babel,  I.  Buch,  from 
the  beginnings  to  the  time  of  Christ,  Minister  i.  Westfalen,  1907.  It 
also  has  numerous  tables  containing  the  above  zero. 

51 


52  THE  HINDU-ARABIC  NUMERALS 

use  of  this  Babylonian  symbol  seems  to  have  been  in  the 
fractions,  60ths,  3600ths,  etc.,  and  somewhat  similar  to 
the  Greek  use  of  o,  for  ovhiv,  with  the  meaning  vacant. 

"The  earliest  undoubted  occurrence  of  a  zero  in  India  is 
an  inscription  at  Gwalior,  dated  Samvat  933  (876  A. d.}. 
Where  50  garlands  are  mentioned  (line  20),  50  is  written 
£]  O.  270  (line  4)  is  written  V?°-" 1  The  Bakhsali  Manu- 
script 2  probably  antedates  this,  using  the  point  or  dot  as 
a  zero  symbol.  Bayley  mentions  a  grant  of  Jaika  Rash- 
trakuta  of  Bharuj,  found  at  Okamandel,  of  date  738  A.D., 
which  contains  a  zero,  and  also  a  coin  with  indistinct 
Gupta  date  707  (897  a.d.),  but  the  reliability  of  Bay- 
ley's  work  is  questioned.  As  has  been  noted,  the  appear- 
ance of  the  numerals  in  inscriptions  and  on  corns  would 
be  of  much  later  occurrence  than  the  origin  and  written 
exposition  of  the  system.  From  the  period  mentioned 
the  spread  was  rapid  over  all  of  India,  save  the  southern 
part,  where  the  Tamil  and  Malayalam  people  retain  the 
old  system  even  to  the  present  day.3 

Aside  from  its  appearance  in  early  inscriptions,  there 
is  still  another  indication  of  the  Hindu  origin  of  the  sym- 
bol in  the  special  treatment  of  the  concept  zero  in  the 
early  works  on  arithmetic.  Brahmagupta,  who  lived  in 
Ujjain,  the  center  of  Indian  astronomy,4  in  the  early  part 

1  From  a  letter  to  D.  E.  Smith,  from  G.  F.  Hill  of  the  British 
Museum.  See  also  his  monograph  "On  the  Early  Use  of  Arabic  Nu- 
merals in  Europe,"  in  Archceologia,  Vol.  LXII  (1910),  p.  137. 

2  R.  Hoernle,  "The  Bakshall  Manuscript,"  Indian  Antiquary,  Vol. 
XVII,  pp.  33-48  and  275-279,  1888 ;  Thibaut,  Astronomic,  Astrulogie 
und  Mathematik,  p.  75 ;   Hoernle,  Verhandlungen,  loc.  cit.,  p.  132. 

3  Bayley,  loc.  cit.,  Vol.  XV,  p.  29.  Also  Bendall,  "On  a  System  of 
Numerals  used  in  South  India,"  Journal  of  the  Iioyal  Asiatic  Society, 
1896,  pp.  789-792. 

4  V.  A.  Smith,  The  Early  History  of  India,  2d  ed.,  Oxford,  1908, 
p.  14. 


THE  SYMBOL  ZERO  53 

of  the  seventh  century,  gives  in  his  arithmetic  1  a  distinct 
treatment  of  the  properties  of  zero.  He  does  not  discuss 
a  symbol,  but  he  shows  by  his  treatment  that  in  some 
way  zero  had  acquired  a  special  significance  not  found  in 
the  Greek  or  other  ancient  arithmetics.  A  still  more 
scientific  treatment  is  given  by  Bhaskara,2  although  in 
one  place  he  permits  himself  an  unallowed  liberty  in 
dividing  by  zero.  The  most  recently  discovered  work 
of  ancient  Indian  mathematical  lore,  the  Ganita-Sara- 
Sahgraha3  of  Mahaviracarya  (c.  830  a.d.),  while  it  does 
not  use  the  numerals  with  place  value,  has  a  similar  dis- 
cussion of  the  calculation  with  zero. 

What  suggested  the  form  for  the  zero  is,  of  course, 
purely  a  matter  of  conjecture.  The  dot,  which  the  Hin- 
dus used  to  fill  up  lacunae  in  then  manuscripts,  much  as 
we  indicate  a  break  in  a  sentence,4  would  have  been  a 
more  natural  symbol ;  and  this  is  the  one  which  the  Hin- 
dus first  used  5  and  which  most  Arabs  use  to-day.  There 
was  also  used  for  this  purpose  a  cross,  like  our  X,  and  this 
is  occasionally  found  as  a  zero  symbol.6  In  the  Bakhsali 
manuscript  above  mentioned,  the  word  sunya,  with  the 
dot  as  its  symbol,  is  used  to  denote  the  unknown  quan- 
tity, as  well  as  to  denote  zero.    An  analogous  use  of  the 

1  Colebrooke,  Algebra,  with  Arithmetic  and  Mensuration,  from  the 
Sanskrit  of  Brahmegupta  and  Bhdscara,  London,  1817,  pp.  339-340. 

2  Ibid.,  p.  138. 

3  D.  E.  Smith,  in  the  Bibliotheca  Mathematica,  Vol.  IX  (3),  pp.  106- 
110. 

4  As  when  we  use  three  dots  (...). 

5  "The  Hindus  call  the  nought  explicitly  sunyabindu  'the  dot 
marking  a  blank,1  and  about  500  a.d.  they  marked  it  by  a  simple  dot, 
which  latter  is  commonly  used  in  inscriptions  and  MSS.  in  order  to 
mark  a  blank,  and  which  was  later  converted  into  a  small  circle." 
[Biihler,  On  the  Origin  of  the  Indian  Alphabet,  p.  53,  note.] 

6  Fazzari,  DelV  origine  delle  parole  zero  e  cifra,  Naples,  1903. 


54  THE  HINDU-ARABIC  NUMERALS 

zero,  for  the  unknown  quantity  in  a  proportion,  appears 
in  a  Latin  manuscript  of  some  lectures  by  Gottfried 
Wolack  in  the  University  of  Erfurt  in  1467  and  1468.1 
The  usage  was  noted  even  as  early  as  the  eighteenth 
century.2 

The  small  circle  was  possibly  suggested  by  the  spurred 
circle  which  was  used  for  ten.3  It  has  also  been  thought 
that  the  omicron  used  by  Ptolemy  in  his  Almagest,  to 
mark  accidental  blanks  in  the  sexagesimal  system  which  he 
employed,  may  have  influenced  the  Indian  writers.4  This 
symbol  was  used  quite  generally  in  Europe  and  Asia,  and 
the  Arabic  astronomer  Al-BattanI5  (died  929  a.d.)  used 
a  similar  symbol  in  connection  with  the  alphabetic  system 
of  numerals.  The  occasional  use  by  Al-BattanI  of  the 
Arabic  negative,  la,  to  indicate  the  absence  of  minutes 

1  E.  Wappler,  "Zur  Geschichte  der  Mathematik  im  15.  Jahrhun- 
dert,"  in  the  Zeitschrift  fiir  Mathematik  und  Physik,  Vol.  XLV,  Hist.- 
lit.  AM.,  p.  47.   The  manuscript  is  No.  C.  80,  in  the  Dresden  library. 

2  J.  G.  Prandel,  Algebra  nebst  Hirer  literarischen  Geschichte,  p.  572, 
Munich,  1795. 

3  See  the  table,  p.  23.  Does  the  fact  that  the  early  European  arith- 
metics, following  the  Arab  custom,  always  put  the  0  after  the  9,  sug- 
gest that  the  0  was  derived  from  the  old  Hindu  symbol  for  10  ? 

4  Bayley,  loc.  cit.,  p.  48.  From  this  fact  Delambre  (Histoire  de  Vas- 
tronomle  ancienne)  inferred  that  Ptolemy  knew  the  zero,  a  theory 
accepted  by  Chasles,  Apercu  historique  sur  Vorigine  et  le  developpement 
des  mlthodes  en ge'ome'trie,  1875 ed.,  p. 476;  Nesselmann,  however,  showed 
(Algebra  der  Griechen,  1842,  p.  138),  that  Ptolemy  merely  used  o  for 
ovdip,  with  no  notion  of  zero.  See  also  G.  Eazzari,  "DelP  origine  delle 
parole  zero  e  cifra,"  Ateneo,  Anno  I,  No.  11,  reprinted  at  Naples  in 
1903,  where  the  use  of  the  point  and  the  small  cross  for  zero  is  also 
mentioned.  Th.  H.  Martin,  Les  signes  numeraux  etc.,  reprint  p.  30,  and 
J.  Brandis,  Das  Miinz-,  Mass-  und  Gewichtswesen  in  Yorderasien  bis  auf 
Alexander  den  Grossen,  Berlin,  1866,  p.  10,  also  discuss  this  usage  of  o, 
without  the  notion  of  place  value,  by  the  Greeks. 

5  Al-Battani  sive  Albatenii  opus  astronomic  urn.  Ad  fidem  codicis 
escurialensis  arabice  editum,  latine  versum,  adnotationibus  instructum 
a  Carolo  Alphonso  Nallino,  1800-1907.  Publicazioni  del  K.Osserva- 
torio  di  Brera  in  Milano,  No.  XL. 


THE  SYMBOL  ZERO  55 

(or  seconds),  is  noted  by  Nallino.1  Noteworthy  is  also 
the  use  of  the  o  for  unity  in  the  Sarada  characters  of  the 
Kashmirian  Atharva-Veda,  the  writing  being  at  least  400 
years  old.  Bhaskara  (c.  1150)  used  a  small  circle  above 
a  number  to  indicate  subtraction,  and  in  the  Tartar  writ- 
ing a  redundant  word  is  removed  by  drawing  an  oval 
around  it.  It  would  be  interesting  to  know  whether  our 
score  mark  (5),  read  "  four  in  the  hole,"  could  trace  its 
pedigree  to  the  same  sources.  O'Creat2  (c.  1130),  in  a 
letter  to  his  teacher,  Adelhard  of  Bath,  uses  r  for  zero, 
being  an  abbreviation  for  the  word  teca  which  we  shall 
see  was  one  of  the  names  used  for  zero,  although  it  could 
quite  as  well  be  from  r^typa.  More  rarely  O'Creat  uses 
O,  applying  the  name  cyfra  to  both  forms.  Frater  Sigs- 
boto3  (c.  1150)  uses  the  same  symbol.  Other  peculiar 
forms  are  noted  by  Heiberg  4  as  being  in  use  among  the 
Byzantine  Greeks  in  the  fifteenth  century.  It  is  evident 
from  the  text  that  some  of  these  writers  did  not  under- 
stand the  import  of  the  new  system.5 

Although  the  dot  was  used  at  first  in  India,  as  noted 
above,  the  small  circle  later  replaced  it  and  continues  in 
use  to  this  day.    The  Arabs,  however,  did  not  adopt  the 

1  Loc.  cit.,  Vol.  II,  p.  271. 

2  C.  Henry,  "Prologus  N.  Ocreati  in  Helceph  ad  Adelardnm  Baten- 
seni  magistrum  snum,"  Ahhandlungen  zur  Geschichte  der  Mathematik, 
Vol.  Ill,  1880. 

3  Max.  Curtze,  "Ueber  eine  Algorismus-Schrift  des  XII.  Jahrhun- 
derts,"  Ahhandlungen  zur  Geschichte  der  Mathematik,  Vol.  VIII,  1898, 
pp.  1-27  ;  Alfred  Nagl,  "  Ueber  eine  Algorismus-Schrift  des  XII.  Jahr- 
hunderts  und  iiber  die  Verbreitung  der  indisch-arabischen  Rechenkunst 
und  Zahlzeichen  im  christl.  Abendlande,"  Zeitschrift  fur  Mathematik 
und  Physik,  Hist.-lit.  Abth.,  Vol.  XXXIV,  pp.  129-146  and  161-170, 
with  one  plate. 

4  "Byzantinische  Analekten,"  Ahhandlungen  zur  Geschichte  der 
Mathematik,  Vol.  IX.  pp.  161-189. 

5  U  org  for  0.    H  also  used  for  5.    |  \XJm  for  13.    [Heiberg,  loc.  cit.] 


\ 


56  THE  HINDU-ARABIC  NUMERALS 

circle,  since  it  bore  some  resemblance  to  the  letter  which 
expressed  the  number  five  in  the  alphabet  system.1  The 
earliest  Arabic  zero  known  is  the  clot,  used  in  a  manu- 
script of  873  a.d.2  Sometimes  both  the  dot  and  the  circle 
are  used  in  the  same  work,  having  the  same  meaning, 
which  is  the  case  in  an  Arabic  MS.,  an  abridged  arith- 
metic of  Jamshid,3  982  a.h.  (1575  a.d.).  As  given  in 
this  work  the  numerals  are  ^  AV^  &  )*  I0?!?.  The  form 
for  5  varies,  in  some  works  becoming  <P  or  co ;  O  is 
found  in  Egypt  and  fc  appears  in  some  fonts  of  type. 
To-day  the  Arabs  use  the  0  only  when,  under  European 
influence,  they  adopt  the  ordinary  system.  Among  the 
Chinese  the  first  definite  trace  of  zero  is  in  the  work  of 
Tsin4  of  1247  a.d.  The  form  is  the  circular  one  of  the 
Hindus,  and  undoubtedly  was  brought  to  China  by  some 
traveler.  *S 

The  name  of  this  all-important  symbol  also  demands 
some  attention,  especially  as  we  are  evert  yet  quite  un- 
decided as  to  what  to  call  it.  We  speak  of  it  to-day  as 
zero,  naught,  and  even  cipher;  the  telephone  operator 
often  calls  it  0,  and  the  illiterate  or  careless  person  calls 
it  aught.  In  view  of  all  this  uncertainty  we  may  well 
inquire  what  it  has  been  called  in  the  past.5 

1  Gerhardt,  Etudes  historiques  sur  V arithmetique  de  position,  Berlin, 
1850,  p.  12 ;  J.  Bowring,  The  Decimal  System  in  Numbers,  Coins,  &  Ac- 
counts, London,  1854,  p.  33. 

2  Karabaj»ek,  Wiener  Zeitschrift  fur  die  Kunde  des  Morgerdandes, 
Vol.  XI,  p.  13  ;  Fiihrer  durch  die  Papyrus-Ausstellung  Erzherzog  Rainer, 
Vienna,  1894,  p.  216. 

3  In  the  library  of  G.  A.  Plimpton,  Esq. 

*  Cantor,  Geschichte,  Vol.  I  (3),  p.  074;  Y.  Mikami,  "A  Remark  on 
the  Chinese  Mathematics  in  Cantor's  Geschichte  der  Mathematik," 
Archiv  der  Mathematik  und  Physik,  Vol.  XV  (8),  pp.  68-70. 

5  Of  course  the  earlier  historians  made  innumerable  guesses  as  to 
the  origin  of  the  word  cipher.    E.g.  Matthew  Hostus,  Be  numeratione 


THE  SYMBOL  ZERO  57 

As  already  stated,  the  Hindus  called  it  sunya,  "void."  1 
This  passed  over  into  the  Arabic  as  as-sifr  or  sift:2.  When 
Leonard  of  Pisa  (1202)  wrote  upon  the  Hindu  numerals 
he  spoke  of  this  character  as  zephirum.3  Maximus  P la- 
nudes  (1330),  writing  under  both  the  Greek  and  the  Ara- 
bic influence,  called  it  tziphra.4  In  a  treatise  on  arithmetic 
written  in  the  Italian  language  by  Jacob  of  Florence5 

emendata,  Antwerp,  1582,  p.  10,  says:  " Siphra  vox  Hebrpeam  originem 
sapit  ref  grtque  :  &  ut  clocti  arbitrantur,  a  verbo  saphar,  quod  Ordine 
numerauit  signincat.  Unde  Sephar  numerus  est :  bine  Siphra  (vulgo 
corruptius) .  Etsi  vero  gens  Iudaica  his  notis,  quse  bodie  Siphrse 
vocantur,  usa  non  fuit :  mansit  tamen  rei  appellatio  apud  multas 
gentes."  Dasypodius,  Institutiones  malhematicae,  Vol.  I,  1593,  gives  a 
large  part  of  this  quotation  word  for  word,  without  any  mention  of 
the  source.  Herinannus  Hugo,  De  prima  scribendi  origine,  Trajecti  ad 
Rhenum,  1738,  pp.  304-305,  and  note,  p.  305;  Karl  Krumbacber, 
"Woher  stainmt  das  Wort  Ziffer  (Chiffre)  ?",  Etudes  de  philologie 
neo-grecque,  Paris,  1892. 

1  Buhler,  loc.  cit.,  p.  78  and  p.  86. 

2  Fazzari,  loc.  cit.,  p.  4.  So  Elia  Misrachi  (1455-1526)  in  his  post- 
humous Book  of  Number,  Constantinople,  1534,  explains  sifra  as  being 
Arabic.  See  also  Steinschneider,  Bibliotheca  Mathematica,  1893,  p.  69, 
and  G.  Wertheim,  Die  Arithmetik  des  Elia  Misrachi,  Programm,  Frank- 
furt, 1893. 

3  "Cum  his  novem  figuris,  et  cum  hoc  signo  0,  quod  arabice  zephi- 
rum appellator,  scribitur  quilibet  numerus." 

4  Ttfrppa,  a  form  also  used  by  Neophytos  (date  unknown,  probably 
c.  1330).  It  is  curious  that  Finaeus  (1555  ed.,  f.  2)  used  the  form  tzi- 
phra throughout.  A.  J.  H.Vincent  ["Sur  Porigine  de  nos  chiffres," 
Notices  et  Extraits  des  MSS.,  Paris,  1847,  pp.  143-150]  says:  "  Ce  cercle 
f  ut  nomine"  par  les  uns,  sijws,  rota,  galgal  .  .  . ;  par  les  autres  tsiphra 
(de  "IB5J,  couronne  ou  diademe)  ou  ciphra  (de  "ICE,  numeration)."  Cb. 
de  Paravey,  Essai  sur  V origine  unique  et  hieroglyphique  des  chiffres  et  des 
lettres  de  tous  les  peuples,  Paris,  1826,  p.  165,  a  rather  fanciful  work, 
gives  "  vase,  vase  arrondi  et  ferm6  par  un  couvercle,  qui  est  le  symbole 
de  la  10e  Heure,  J,"  among  the  Chinese  ;  also  "Tsiphron  Zdron,  ou 
tout  a  fait  vide  en  arabe,  rft'^pa  en  grec  .  .  .  d'oii  chiffre  (qui  derive 
plutot,  suivant  nous,  de  TH^breu  Sepher,  compter.") 

5  "Compilatus  a  Magistro  Jacobo  de  Florentia  apud  montem  pesa- 
lanum,"  and  described  by  G.  Lami  in  his  Catalogus  codicum  manu- 
scriptorum  qui  in  bibliotheca  Riccardiana  Florent'm  adservantur.  See 
Fazzari,  loc.  cit.,  p.  5. 


58  THE  HINDLT-ARABIC  NUMERALS 

(1307)  it  is  called  zeuero,1  while  in  an  arithmetic  of  Gio- 
vanni di  Danti  of  Arezzo  (1370)  the  word  appears  as 
feuero.2  Another  form  is  zepiro,3  which  was  also  a  step 
from  zephirum  to  zero.4 

Of  course  the  English  cipher,  French  chiffre,  is  derived 
from  the  same  Arabic  word,  as-sifr,  but  in  several  lan- 
guages it  has  come  to  mean  the  numeral  figures  in  general. 
A  trace  of  this  appears  hi  our  word  ciphering,  meaning 
figuring  or  computing.5  Johann  Huswirt6  uses  the  word 
with  both  meanings ;  he  gives  for  the  tenth  character 
the  four  names  theca,  cir -cuius,  cifra,  and  figura  nihili. 
In  this  statement  Huswirt  probably  follows,  as  did  many 
writers  of  that  period,  the  Algorismus  of  Johannes  de 
Sacrobosco  (c.  1250  a.d.),  who  was  also  known  as  John 
of  Halifax  or  John  of  Holywood.    The  commentary  of 

1  "Et  doveto  sapere  chel  zeuero  per  se  solo  non  significa  nulla  ma 
e  potentia  di  fare  significare,  .  .  .  Et  decina  o  centinaia  o  migliaia 
non  si  puote  scrivere  senza  questo  segno  0.  la  quale  si  chiama  zeuero." 
[Fazzari,  loc.  cit.,  p.  5.] 

2  Ibid.,  p.  6. 

3  Avicenna  (980-1036),  translation  by  Gasbarri  et  Frangois,  "piu  il 
punto  (gli  Arabi  adoperavano  il  punto  in  vece  dello  zero  il  cui  segno 
0  in  arabo  si  chiama  zepiro  donde  il  vocabolo  zero),  che  per  se  stesso 
non  esprime  nessun  numero."  This  quotation  is  taken  from  D.C. 
Martines,  Origine  e  progressi  deW  aritmetica,  Messina,  1865. 

4  Leo  Jordan,  "Materialien  zur  Geschichte  der  arabischen  Zahl- 
zeichen  in  Frankreich,"  Archiv  fur  Kulturgeschlchte,  Berlin,  1905, 
pp.  155-195,  gives  the  following  two  schemes  of  derivation,  (1)  "zefiro, 
zeviro,  zeiro,  zero,"  (2)  "zefiro,  zefro,  zevro,  zero." 

5  Kobel  (1518  ed.,  f.  A4)  speaks  of  the  numerals  in  general  as  "die 
der  gemain  man  Zyfer  nendt."  Recorde  (Grounde  of  Artes,  1558  ed., 
f.  I50)  says  that  the  zero  is  "called  priuatly  a  Cyphar,  though  all  the 
other  sometimes  be  likewise  named." 

6  "Decimo  X0  theca,  circul?  cifra  sive  figura  nihili  appelat'." 
[Enchiridion  Algorismi,  Cologne,  1501.]  Later,  "quoniam  de  integris 
tarn  in  cifris  quam  in  pjroiectilibus," — the  won!  proiectUUms  referring 
to  markers  "thrown"  and  ased  on  an  abacus,  whence  the  French 
jetons  and  the  English  expression  "to  cast  an  account." 


THE  SYMBOL  ZERO  59 

Petrus  de  Dacia1  (c.  1291  a.d.)  on  the  Algorismus  vul- 
garis of  Sacrobosco  was  also  widely  used.  The  wide- 
spread use  of  this  Englishman's  work  on  arithmetic  in 
the  universities  of  that  time  is  attested  by  the  large  num- 
ber2 of  MSS.  from  the  thirteenth  to  the  seventeenth  cen- 
tury still  extant,  twenty  in  Munich,  twelve  in  Vienna, 
thirteen  in  Erfurt,  several  in  England  given  by  Halli- 
well,3  ten  listed  in  Coxe's  Catalogue  of  the  Oxford  College 
Library,  one  in  the  Plimpton  collection,4  one  in  the 
Columbia  University  Library,  and,  of  course,  many 
others. 

From  as-sifr  has  come  zephyr,  cipher,  and  finally  the 
abridged  form  zero.  The  earliest  printed  work  in  which 
is  found  this  final  form  appears  to  be  Calandri's  arith- 
metic of  1491,5  while  in  manuscript  it  appears  at  least  as 
early  as  the  middle  of  the  fourteenth  century.6  It  also 
appears  in  a  work,  Le  Kadran  des  marchans,  by  Jehan 


1  "  Decima  vero  o  dicitur  teca,  circulus,  vel  cyf  ra  vel  figura  nichili." 
[Maximilian  Curtze,  Petri  Philomeni  de  Dacia  in  Algorismum  Vulga- 
rem  Johannis  de  Sacrobosco  commentarius,  una  cum  Algorismo  ipso, 
Copenhagen,  1897,  p.  2.]  Curtze  cites  five  manuscripts  (fourteenth 
and  fifteenth  centuries)  of  Dacia's  commentary  in  the  libraries  at 
Erfurt,  Leipzig,  and  Salzburg,  in  addition  to  those  given  by  Enestrom, 
Ofversigt  af  Kongl.  Vetenskaps-Akademiens  Forhandlingar,  1885,  pp. 
15-27,  65-70 ;  1886,  pp.  57-60. 

2  Curtze,  loc.  cit.,  p.  vi. 

8  Rara  Mathematica,  London,  1841,  chap,  i,  "Joannis  de  Sacro- 
Bosco  Tractatus  de  Arte  Numerandi." 

4  Smith,  Eara  Arithmetica,  Boston,  1909. 

5  In  the  1484  edition,  Borghi  uses  the  form  "cefiro  :  ouero  nulla  :  " 
while  in  the  1488  edition  he  uses  "zefiro:  ouero  nulla,"  and  in  the 
1540  edition,  f.  3,  appears  "Chiamata  zero,  ouero  nulla."  Woepcke 
asserted  that  it  first  appeared  in  Calandri  (1491)  in  this  sentence  : 
"Sono  dieci  le  figure  con  le  quali  ciascuno  numero  si  puo  significarc  : 
delle  quali  n'e  una  che  si  chiama  zero :  et  per  se  sola  nulla  significa" 
(f.  4).    [See  Propagation,  p.  522.] 

6  Boncompagni  Bullttino,  Vol.  XVI,  pp.  673-685. 


60  THE  HINDU-ARABIC  NUMERALS 

Certain,1  written  in  1485.  This  word  soon  became  fairly 
well  known  in  Spain  2  and  France.3  The  medieval  writers 
also  spoke  of  it  as  the  sipos,*  and  occasionally  as  the 
wheel,5   cireulus6    (in    German   das  Ringlein1},   circular 

1  Leo  Jordan,  loc.  cit.  In  the  Catalogue  of  MSS.,  Bibl.  de  V Arsenal, 
Vol.  Ill,  pp.  154-155,  this  wqrk  is  No.  2904  (184  S.A.F.),  Bibl.  Nat., 
and  is  also  called  Petit  traicte  de  algorisme. 

2  Texada  (1546)  says  that  there  are  "nueue  letros  yvn  zero  o  cifra  " 
(i.  3). 

3  Savonne  (1563,  1751  ed.,  f.  1):  "Vne  ansi  formee  (o)  qui  s'appelle 
nulle,  &  entre  rnarchans  zero,"  showing  the  influence  of  Italian  names 
on  French  mercantile  customs.  Trenchant  (Lyons,  1566,  1578  ed.,  p. 
12)  also  says :  "  La  derniere  qui  s'apele  nulle,  ou  zero  ; "  but  Champe- 
nois,  his  contemporary,  writing  in  Paris  in  1577  (although  the  work 
was  not  published  until  1578),  uses  "cipher,"  the  Italian  influence 
showing  itself  less  in  this  center  of  university  culture  than  in  the  com- 
mercial atmosphere  of  Lyons. 

4  Thus  Radulph  of  Laon  (c.  1100):  "Inscribitur  in  ultimo  ordine  et 

figura    \»J  sipos  nomine,  quae,  licet  numerum  nullum  signitet,  tan- 

tum  ad  alia  quaedam  utilis,  ut  insequentibus  declarabitur."  ["Der 
Arithmetische  Tractat  des  Radulph  von  Laon,"  Abhandlungen  zur  Ge- 
schichte  der  Mathematik,  Vol.  V,  p.  97,  from  a  manuscript  of  the  thir- 
teenth century.]  Chasles  (Comptes  rendus,  t.  16,  1843,  pp.  1393,  1408) 
calls  attention  to  the  fact  that  Radulph  did  not  know  how  to  use  the 
zero,  and  he  doubts  if  the  sipos  was  really  identical  with  it.    Radulph 

says:  ".  .  .  figuram,  cui  sipos  nomen  est  (  •  j  in  motum  rotulae  f or- 

matam  nullius  numeri  significatione  inscribi  solere  praediximus,"  and 
thereafter  uses  rotula.  He  uses  the  sipos  simply  as  a  kind  of  marker 
on  the  abacus. 

5  Rabbi  ben  Ezra  (1092-1168)  used  both  bib),  galgal  (the  Hebrew  for 
wheel),  and  NICE,  sifra.  See  M.  Steinschneider,  "  Die  Mathematik  bei 
den  Juden,"  in  Bibliotheca  Mathematica,  1893,  p.  69,  and  Silberberg, 
Das  Buch  der  Zahl  des  B.  Abraham  ibn  Esra,  Frankfurt  a.  M.,  1895,  p. 
96,  note  23  ;  in  this  work  the  Hebrew  letters  are  used  for  numerals  with 
place  value,  having  the  zero. 

6  E.g.,  in  the  twelfth-century  Liber  algorismi  (see  Boncompagni's 
Trattati,  II,  p.  28).  So  Ramus  (Libri  II,  1569  ed.,  p.  1)  says:  "Cir- 
culus  quse  nota  est  ultima  :  nil  per  se  significat."  (See  also  the  Scho- 
nerus  ed.  of  Ramus,  1586,  p.  1.) 

7  "Und  wirt,  das  ringlein  o.  die  Ziffer  genant  die  nichts  bedeut." 
[KobePs  L'txhcnbuch,  1549  ed.,  f.  10,  and  other  editions.] 


THE  SYMBOL  ZERO  61 

7iote,1  theca,2  long  supposed  to  be  from  its  resemblance  to 
the  Greek  theta,  but  explained  by  Petrus  de  Daciaas  being 
derived  from  the  name  of  the  iron  3  used  to  brand  thieves 
and  robbers  with  a  circular  mark  placed  on  the  forehead  or 
on  the  cheek.  It  was  also  called  omicron 4  (the  Greek  o), 
being  sometimes  written  o  or  <j>  to  distinguish  it  from  the 
letter  o.  It  also  went  by  the  name  null 5  (in  the  Latin  books 

1  I.e.  "circular  figure,"  our  word  notation  having  come  from  the 
medieval  nota.  Thus  Tzwivel  (1507,  f.  2)  says:  "Nota  autem  circula- 
ris  .o.  per  se  sumpta  nihil  vsus  habet.  alijs  tamen  adiuncta  earum 
significantiam  et  auget  et  ordinem  permutat  quantum  quo  ponit  ordi- 
nein.  vt  adiuncta  note  binarij  hoc  modo  20  facit  earn  significare  bis 
decern  etc."  Also  (ibid.,  f.  4),  "figura  circularis,"  "circularis  nota." 
Clichtoveus  (1503  ed.,  f.  xxxvn)  calls  it  "nota  aut  circularis  o," 
"circidaris  nota,"  and  "figura  circularis."  Tonstall  (1522,  f.  B3)  says 
of  it:  "Decimo  uero  nota  ad  formam  .O-  litterae  circulari  figura  est: 
quam  alij  circulum,  uulgus  cyphram  uocat,"  and  later  (f .  C4)  speaks 
of  the  "circulos."  Grammateus,  in  his  A Igorismus  de  integris  (Erfurt, 
1523,  f .  A2),  speaking  of  the  nine  significant  figures,  remarks :  "  His  au- 
tem superadditur  decima  figura  circularis  ut  0  existens  que  ratione  sua 
nihil  significat."  Noviomagus  (De  Numeris  libri  II,  Paris,  1539,  chap. 
xvi,  "De  notis  numerorum,  quas  zyphras  vocant")  calls  it  "circularis 
nota,  quam  ex  his  solam,  alij  sipheram,  Georgius  Valla  zyphram." 

2  Huswirt,  as  above.  Ramus  (Scholae  mathematicae,  1569  ed.,  p.  112) 
discusses  the  name  interestingly,  saying:  "Circulum  appellamus  cum 
multis,  quam  alii  thecam,  alii  figuram  nihili,  alii  figuram  privationis, 
sen  figuram  nullam  vocant,  alii  ciphram,  cum  tamen  hodie  omnes  hse 
notae  vulgo  ciphrse  nominentur,  &  his  notis  numerare  idem  sit  quod 
ciphrare."  Tartaglia  (1592  ed.,  f.  9)  says:  "si  chiama  da  alcuni  tecca, 
da  alcuni  circolo,  da  altri  cifra,  da  altri  zero,  &  da  alcuni  altri  nulla." 

3  "  Quare  autem  aliis  nominibus  vocetur,  non  dicit  auctor,  quia 
omnia  alia  nomina  habent  rationem  suae  lineationis  sive  figurationis. 
Quia  rotunda  est,  dicitur  haec  figura  teca  ad  similitudinem  tecae. 
Teca  enim  est  ferrum  figurae  rotundae,  quod  ignitum  solet  in  quibus- 
dam  regionibus  imprimi  fronti  vel  maxillae  furis  seu  latronum."  [Loc. 
cit.,  p.  20.]  But  in  Greek  theca  (0HKH,  017/07)  is  a  place  to  put  some- 
thing, a  receptacle.  If  a  vacant  column,  e.g.  in  the  abacus,  was  so 
called,  the  initial  might  have  given  the  early  forms  ©  and  0  for  the  zero. 

4  Buteo,  Logistica,  Lyons,  1559.  See  also  Wertheim  in  the  Biblio- 
theca  Mathematical  1901,  p.  214. 

5  "  O  est  appellee  chiffre  ou  nulle  ou  figure  de  nulle  valeur."  [La 
Roche,  L'arithmetique,  Lyons,  1520.] 


62  THE  HINDU-ARABIC  NUMERALS 

nihil1  or  nulla,2  and  in  the  French  Wen3),  and  very  com- 
nii  >nly  by  the  name  cipher.4  Walli.s  5  gives  one  of  the  earli- 
est extended  discussions  of  the  various  forms  of  the  word, 
giving  certain  other  variations  worthy  of  note,  as  ziphra,  zi- 
fera,  siphra,  eiphra,  tsiphra,  tziphra,  and  the  Greek  r^icf)pa.G 

1  "  Decima  autem  figura  nihil  uocata,"  "  figura  nihili  (quam  etiam 
cifram  uocant).""    [Stifel,  Arithmetica  Integra,  1544,  f.  1.] 

2  "  Zifra,  &  Nulla  uel  figura  Nihili."  [Scheubel,  1545,  p.  1  of  ch.  1.] 
Nulla  is  also  used  by  Italian  writers.  Thus  Sfortunati  (1545  ed.,  f .  4) 
says :  "  et  la  decima  nulla  &  e  chiamata  questa  decima  zero  ; "  Cataldi 
(1602,  p.  1):  "La  prima,  che  e  o,  si  chiama  nulla,  ouero  zero,  ouero 
niente."  It  also  found  its  way  into  the  Dutch  arithmetics,  e.g.  Raets 
(1576,  1580  ed.,  f .  A3):  »  Nullo  dat  ist  niet ;"  Van  der  Schuere  (1600, 
1624  ed.,  f.  7);  Wilkens  (1669  ed.,  p.  1).  In  Germany  Johann  Albert 
(Wittenberg,  1534)  and  Rudolff  (1526)  both  adopted  the  Italian  nulla 
and  popularized  it.  (See  also  Kuckuck,  Die  Eechenkunst  im  sechzehn- 
ten  Jahrhundert,  Berlin,  1874,  p.  7  ;  Giinther,  Geschichte,  p.  316.) 

3  "La  dixieme  s'appelle  chifre  vulgairement :  les  vns  1' appellant 
zero:  nous  la  pourrons  appeller vn Rien."    [Peletier,  1607  ed.,  p.  14.] 

4  It  appears  in  the  Polish  arithmetic  of  Klos  (1538)  as  cyfra.  "The 
Ciphra  0  augmenteth  places,  but  of  himselfe  signifieth  not,"  Digges, 
157!),  p.  1.  Hodder  (10th  ed.,  1672,  p.  2)  uses  only  this  word  (cypher 
or  cipher),  and  the  same  is  true  of  the  first  native  American  arithme- 
tic, written  by  Isaac  Greenwood  (1720,  p.  1).  Petrus  de  Dacia  derives 
cyfra  from  circumference.  "Vocatur  etiam  cyfra,  quasi  circumfacta 
vel  circumferenda,  quod  idem  est,  quod  circulus  non  habito  respectu 
ad  centrum."    [Loc.  cit.,  p.  26.] 

5  Opera  mathematica,  1605,  Oxford,  Vol.  I,  chap,  ix,  Mathesis  univer- 
salis, "De  figuris  numeralibus,"  pp.  46-49;  Vol.  II,  Algebra,  p.  10. 

6  Martin,  Origine  de  notre  systeme  de  numeration  icrite,  note  140,  p.  36 
of  reprint,  spells  ralcj>pa  from  Maximus  Planudes,  citing  Wallis  as  an 
autbority.   This  is  an  error,  for  Wallis  gives  the  correct  form  as  above. 

Alexander  von  Humboldt,  "Uber  die  bei  verscbiedenen  VOlkern 
iiblichen  Systeme  von  Zahlzeichen  und  uber  den  Ursprung  des  Stellen- 
werthes  in  den  indiscken  Zahlen,"  Crelle's  Journal  fur  reine  und 
angewandte  Mathcmatik,  Vol.  IV,  1829,  called  attention  to  the  work 
api6/j.ol'li>8iKoi  of  the  monk  Neophytos,  supposed  to  be  of  the  four- 
teenth century.  In  this  work  the  forms  T^xppa  and  Tttffjuppa  appear. 
See  also  Boeckh,  De  abaco  Graccorum,  Berlin,  1841,  and  Tannery,  "Le 
Scholie  du  moine  Neophytos,"  Revue  Archeologique,  1885,  pp.  99-102. 
Jordan,  loc.  cit.,  gives  from  twelfth  and  thirteenth  century  manuscri] its 
the  forms  cifra,  ciffre,  chifras,  and  cifrus.  Du  Cange,  Glossarium  mediae 
et  infimae  Latinitatis,  Paris,  1842,  gives  also  chilerae.  Dasypodius, 
lnrtitutiones  Mul/u  niaticae,  Strasslmn;-,  1503-1506,  adds  the  forms 
zyphra  and  syphra.  Boissiere,  L'urt  d'arythmetique  contenant  loute 
dimention,  tressingulier  ct  commode,  taut  pour  Cart  militaire  <jue  autrcs 
calculations,  Paris,  1551:  "Puis  y  en  a  vn  autre  diet  zero  lequel  ne 
designe  nulle  quantity  par  soy,  ains  seulement  les  loges  vuides," 


CHAPTER  V 

THE  QUESTION  OF  THE  INTRODUCTION  OF  THE 
NUMERALS  INTO  EUROPE   BY  BOETHIUS 

Just  as  we  were  quite  uncertain  as  to  the  origin  of 
the  numeral  forms,  so  too  are  we  uncertain  as  to  the 
time  and  place  of  their  introduction  into  Europe.  There 
are  two  general  theories  as  to  this  introduction.  The  first 
is  that  they  were  carried  by  the  Moors  to  Spain  in  the 
eighth  or  ninth  century,  and  thence  were  transmitted 
to  Christian  Europe,  a  theory  which  will  be  considered 
later.  The  second,  advanced  by  Woepcke,1  is  that  they 
were  not  brought  to  Spain  by  the  Moors,  but  that  they 
were  already  in  Spain  when  the  Arabs  arrived  there,  having 
reached  the  West  through  the  Neo-Pythagoreans.  'There 
are  two  facts  to  support  this  second  theory :  (1)  the  forms 
of  these  numerals  are  characteristic,  differing  materially 
from  those  which  were  brought  by  Leonardo  of  Pisa 
from  Northern  Africa  early  in  the  thirteenth  century 
(before  1202  a.d.)  ;   (2)  they  are  essentially  those  which 

1  Propagation,  pp.27,  234,  442.  Treutlein,  "Das  Rechnen  im  16. 
Jahrhundert,"  Abhandlungen  zur  Geschichte  der  Mathematik,  Vol.  I, 
p.  5,  favors  the  same  view.  It  is  combated  by  many  writers,  e.g.  A.C. 
Burnell,  loc.  cit.,  p.  59.  Long  before  Woepcke,  I.  F.  and  G.I.Weid- 
ler,  De  characteribus  numerorum  vulgaribus  et  eorum  aelatibus,  Witten- 
berg, 1727,  asserted  the  possibility  of  their  introduction  into  Greece 
by  Pythagoras  or  one  of  his  followers  :  "  Potuerunt  autem  ex  oriente, 
uel  ex  phoenicia,  ad  graecos  traduci,  uel  Pythagorae,  uel  eius  discipu- 
lorum  auxilio,  cum  aliquis  co,  proflciendi  in  Uteris  causa,  iter  faceret, 
et  hoc  quoque  inuentum  addisceret." 

63 


04  THE  HINDU-ARABIC  NUMERALS 

tradition  has  so  persistently  assigned  to  Boethius  (c.  500 
A.D.),  and  which  he  would  naturally  have  received,  if 
at  all,  from  these  same  Neo-Pythagoreans  or  from  the 
sources  from  which  they  derived  them.  Furthermore, 
Woepcke  points  out  that  the  Arabs  on  entering  Spain 
(711  A.D.)  would  naturally  have  followed  their  custom 
of  adopting  for  the  computation  of  taxes  the  numerical 
systems  of  the  countries  they  conquered,1  so  that  the 
numerals  brought  from  Spain  to  Italy,  not  having  under- 
gone the  same  modifications  as  those  of  the  Eastern  Arab 
empire,  would  have  differed,  as  they  certainly  did,  from 
those  that  came  through  Bagdad.  The  theory  is  that  the 
Hindu  system,  without  the  zero,  early  reached  Alexan- 
dria (say  450  a.d.),  and  that  the  Neo-Pythagorean  love 
for  the  mysterious  and  especially  for  the  Oriental  led 
to  its  use  as  something  bizarre  and  cabalistic ;  that  it 
was  then  passed  along  the  Mediterranean,  reaching  Boe- 
thius in  Athens  or  in  Rome,  and  to  the  schools  of  Spain, 
being  discovered  in  Africa  and  Spain  by  the  Arabs  even 
before  they  themselves  knew  the  improved  system  with 
the  place  value. 

1  E.g.,  they  adopted  the  Greek  numerals  in  use  in  Damascus  and 
Syria,  and  the  Coptic  in  Egypt.  Theophanes  (758-818  a.d.),  Chrono- 
graphia,  Scriptores  Historiae  Byzantinae,  Vol.  XXXIX,  Bonnae,  1839, 
p.  575,  relates  that  in  699  a.d.  the  caliph  Walld  forhade  the  use  of  the 
Greek  language  in  the  bookkeeping  of  the  treasury  of  the  caliphate, 
but  permitted  the  use  of  the  Greek  alphabetic  numerals,  since  the 
Arabs  had  no  convenient  number  notation  :  /ecu  e/cwXi/cre  ypd(pe<r0ai  'EX- 
\t]vmttI  tovs  dy/xoaLovs  twv  \oyo6ealwv  kuiSikcls,  dXX  ApafiLots  avra  Trapaarj- 
p.aive<r8ai,  X^P'5  T^v  ^Pv(Pwvi  fTei.87]  advvarov  ry  eKelvwv  yXwaarj  p.ovd8a  r/ 
8vd5a  r)  rpidoa  7;  <5ktu>  rjpicrv  77  rpia  ypdcpecrOai  •  8l6  /ecu  eais  crrifi.ep6v  elaiv 
avv  aureus  vordptoi  XpiaTiavol.  The  importance  of  this  contemporaneous 
document  was  pointed  out  by  Martin,  loc.  cit.  Karabacek,  "Die  In- 
volutio  im  arabischen  Schriftwesen,"  Vol.CXXXVof  SitzungsbericMe 
d.  phil.-hist.  Clause  d.  k.  Akad.  d.  Wiss.,  Vienna,  189G,  p.  25,  gives  an 
Arabic  date  of  808  a.d.  in  Greek  letters. 


THE  BOETHIUS  QUESTION  65 

A  recent  theory  set  forth  by  Bubnov  1  also  deserves 
mention,  chiefly  because  of  the  seriousness  of  purpose 
shown  by  this  well-known  writer.  Bubnov  holds  that 
the  forms  first  found  in  Europe  are  derived  from  ancient 
symbols  used  on  the  abacus,  but  that  the  zero  is  of  Hindu 
origin.  This  theory  does  not  seem  tenable,  however,  in 
the  light  of  the  evidence  already  set  forth. 

Two  questions  are  presented  by  Woepcke's  theory : 
(1)  What  was  the  nature  of  these  Spanish  numerals,  and 
how  were  they  made  known  to  Italy?  (2)  Did  Boethius 
know  them  ? 

The  Spanish  forms  of  the  numerals  were  called  the 
huruf  al-gobdr,  the  gobar  or  dust  numerals,  as  distin- 
guished from  the  huruf  al-jumal  or  alphabetic  numer- 
als. Probably  the  latter,  under  the  influence  of  the 
Syrians  or  Jews,2  were  also  used  by  the  Arabs.  The 
significance  of  the  term  gobar  is  doubtless  that  these 
numerals  were  written  on  the  dust  abacus,  this  plan 
being  distinct  from  the  counter  method  of  representing 
numbers.  It  is  also  worthy  of  note  that  Al-Biru.ni  states 
that  the  Hindus  often  performed  numerical  computations 
in  the  sand.  The  term  is  found  as  early  as  c.  950, 
in  the  verses  of  an  anonymous  writer  of  Kan  wan,  in 
Tunis,  in  which  the  author  speaks  of  one  of  his  works 
on  gobar  calculation  ;  3  and,  much  later,  the  Arab  writer 
Abu  Bekr  Mohammed  ibn  'Abdallah,  surnamed  al-Hassar 


1  The  Origin  and  History  of  Our  Numerals  (in  Russian),  Kiev,  1908 ; 
The  Independence  of  European  Arithmetic  (in  Russian),  Kiev. 

2  Woepcke,  loc.  cit.,  pp.  462,  262. 

3  Woex^cke,  loc.  cit.,  p.  240.  Hisah-al-Gobar,  by  an  anonymous 
author,  probably  Abu  Sahl  Dunash  ibn  Tamim,  is  given  by  Stein- 
schneider,  "Die  Mathernatik  bei  den  Juden,"  Bibliotheca  Mathematical 
1895,  p.  26, 


66  THE  HINDU-ARABIC  NUMERALS 

(the  arithmetician),  wrote  a  work  of  which  the  second 
chapter  was  "  On  the  dust  figures."  * 

The  gobar  numerals  themselves  were  first  made  known 
to  modern  scholars  by  Silvestre  de  Sacy,  who  discovered 
them  in  an  Arabic  manuscript  from  the  library  of  the 
ancient  abbey  of  St.-Germain-des-Pres.2  The  system  has 
nine  characters,  but  no  zero.  A  dot  above  a  character 
indicates  tens,  two  dots  hundreds,  and  so  on,  5  meaning 
50,  and  5  meaning  5000.  It  has  been  suggested  that 
possibly  these  dots,  sprinkled  like  dust  above  the  numer- 
als, gave  rise  to  the  word  gohdr?  but  this  is  not  at  all 
probable.  This  system  of  dots  is  found  in  Persia  at  a 
much  later  date  with  numerals  quite  like  the  modern 
Arabic ; 4  but  that  it  was  used  at  all  is  significant,  for  it 
is  hardly  likely  that  the  western  system  would  go  back  to 
Persia,  when  the  perfected  Hindu  one  was  near  at  hand. 

At  first  sight  there  would  seem  to  be  some  reason  for 
believing  that  this  feature  of  the  gobar  system  was  of 

1  Steinschneider  in  the  Abhandlungcn,  Vol.  Ill,  p.  110. 

2  See  his  Grammaire  arabe,  Vol.  I,  Paris,  1810,  plate  VIII ;  Ger- 
hardt,  Etudes,  pp.  9-11,  and  Entstehung  etc.,  p.  8;#  I.  F.  Weidler, 
Spicilegium  observationum  ad  historiam  notarum  numerallum  perti- 
nentium,  Wittenberg,  1755,  speaks  of  the  "figura  cifrarum  Saracenica- 
rum"  as  being  different  from  that  of  the  "  characterum  Boethianorum," 
which  are  similar  to  the  "  vulgar ' '  or  common  numerals ;  see  also  Hum- 
boldt, loc.  cit. 

3  Gerhardt  mentions  it  in  his  Entstehung  etc.,  p.  8 ;  Woepcke,  Pro- 
pagation, states  that  these  numerals  were  used  not  for  calculation,  but 
very  much  as  we  use  Roman  numerals.  These  superposed  dots  are 
found  with  both  forms  of  numerals  (Propagation,  pp.  244-246). 

4  Gerhardt  (EJtudes,  p.  9)  from  a  manuscript  in  the  Bibliotheque 
Nationale.    The  numeral  forms  are  O  A  V  U  O  S  W  V  1 ,  20  being 

indicated  by  U  and  200  by  [).  This  scheme  of  zero  dots  was  also 
adopted  by  the  Byzantine  Greeks,  for  a  manuscript  of  Planudes  in  the 
Bibliotheque  Nationale  has  numbers  like  H-'d  for  8,100,000.000.  See 
Gerhardt,  Etudes,  p.  1!».  Pihan,  Expose-  etc.,  p.  208,  gives  two  forms, 
Asiatic  and  Maghrebian,  of  "Ghobar"  numerals. 


THE  BOETIIIUS  QUESTION  67 

Arabic  origin,  and  that  the  present  zero  of  these  people,1 
the  dot,  was  derived  from  it.  It  was  entirely  natural  that 
the  Semitic  people  generally  should  have  adopted  such  a 
scheme,  since  then  diacritical  marks  would  suggest  it, 
not  to  speak  of  the  possible  influence  of  the  Greek 
accents  in  the  Hellenic  number  system.  When  we  con- 
sider, however,  that  the  dot  is  found  for  zero  in  the 
Bakhsali  manuscript,2  and  that  it  was  used  in  subscript 
form  in  the  Kitab  al-Fihrist 3  in  the  tenth  century,  and  as 
late  as  the  sixteenth  century,4  although  in  this  case  prob- 
ably under  Arabic  influence,  we  are  forced  to  believe  that 
this  form  may  also  have  been  of  Hindu  origin. 

The  fact  seems  to  be  that,  as  already  stated,5  the  Arabs 
did  not  immediately  adopt  the  Hindu  zero,  because  it 
resembled  their  5 ;  they  used  the  superscript  dot  as 
serving  their  purposes  fairly  well ;  they  may,  indeed, 
have  carried  this  to  the  west  and  have  added  it  to  the 
gobar  forms  already  there,  just  as  they  transmitted  it 
to  the  Persians.  Furthermore,  the  Arab  and  Hebrew 
scholars  of  Northern  Africa  in  the  tenth  century  knew 
these  numerals  as  Indian  forms,  for  a  commentary  on 
the  Sefer  Yeslrdh  by  Abu  Sahl  ibn  Tamim  (probably 
composed  at  Kairwan,  c.  950)  speaks  of  "the  Indian 
arithmetic  known  under  the  name  of  gobdr  or  dust  cal- 
culation." 6   All  this  suggests  that  the  Arabs  may  very 

i  See  Chap.  IV. 

2  Possibly  as  early  as  the  third  century  a.d.,  but  probably  of  the 
eighth  or  ninth.    See  Cantor,  I  (3),  p.  598. 

3  Ascribed  by  the  Arabic  writer  to  India. 

4  See  Woepcke's  description  of  a  manuscript  in  the  Chasles  library, 
"Recherches  sur  l'histoire  des  sciences  niath&natiques  chez  les  orien- 
taux,"  Journal  Asiatique,  IV  (5),  1859,  p.  358,  note. 

5  P.  56. 

6  Reinaud,  Memoire  sur  Vlnde,  p.  399.  In  the  fourteenth  century 
one  Sihab  al-Din  wrote  a  work  on  which  a  scholiast  to  the  Bodleian 


68  THE  HINDU-ARABIC  NUMERALS 

likely  have  known  the  gobar  forms  before  the  numerals 
reached  them  again  in  773.1  The  term  "  gobar  numer- 
als "  was  also  used  without  any  reference  to  the  peculiar 
use  of  dots.2  In  this  connection  it  is  worthy  of  mention 
that  the  Algerians  employed  two  different  forms  of 
numerals  in  manuscripts  even  of  the  fourteenth  cen- 
tury,3 and  that  the  Moroccans  of  to-day  employ  the 
European  forms  instead  of  the  present  Arabic. 

The  Indian  use  of  subscript  dots  to  indicate  the  tens, 
hundreds,  thousands,  etc.,  is  established  by  a  passage  in 
the  Kitdb  al-Fihrist4  (987  A. d.)  in  which  the  writer  dis- 
cusses the  written  language  of  the  people  of  India.  Not- 
withstanding the  importance  of  this  reference  for  the 
early  history  of  the  numerals,  it  has  not  been  mentioned 
by  previous  writers  on  this  subject.  The  numeral  forms 
given  are  those  which  have  usually  been  called  Indian,5 
in  opposition  to  gobar.  In  this  document  the  dots  are 
placed  below  the  characters,  instead  of  being  superposed 
as  described  above.    The  significance  was  the  same. 

In  form  these  gobar  numerals  resemble  our  own  much 
more  closely  than  the  Arab  numerals  do.  They  varied 
more  or  less,  but  were  substantially  as  follows : 

manuscript  remarks:  "The  science  is  called  Algobar  because  the 
inventor  had  the  habit  of  writing  the  figures  on  a  tablet  covered  with 
sand."    [Gerhardt,  Etudes,  p.  11,  note.] 

1  Gerhardt,  Entstehung  etc.,  p.  20. 

2  H.  Suter,  "Das  Rechenbuch  des  Abu  Zakarija  el-Hassar,"  Bibli- 
otheca  Mathematica,  Vol.  II  (3),  p.  15. 

3  A.  Devoulx,  "Les  chiffres  arabes,"  Bevue  Africainc,Vo\.  XVI,  pp. 
455-458. 

4  Kitab  al-Fihrist,  G.  Fliigel,  Leipzig,  Vol.  I,  1871,  and  Vol.  II, 
1872.  This  work  was  published  after  Professor  FliigeFs  death  by  J. 
Roediger  and  A.  Mueller.  The  first  volume  contains  the  Arabic  text 
and  the  second  volume  contains  critical  notes  upon  it. 

5  Like  those  of  line  5  in  the  illustration  on  page  69. 


THE  BOETHIUS  QUESTION  69 

1   j  x  n   t  j  r*  ■*  *>  t 

!°;  ■/   V   1   b  /  |  I  I 

«  ;  *  ?  &  ?  }  *  i) 

The  question  of  the  possible  influence  of  the  Egyptian 
demotic  and  hieratic  ordinal  forms  has  been  so  often 
Suersested  that  it  seems  well  to  introduce  them  at  this 
point,  for  comparison  with  the  gobar  forms.  They  would 
as  appropriately  be  used  in  connection  with  the  Hindu 
forms,  and  the  evidence  of  a  relation  of  the  first  three 
with  all  these  systems  is  apparent.  The  only  further 
resemblance  is  in  the  Demotic  4  and  in  the  9,  so  that  the 
statement  that  the  Hindu  forms  in  general  came  from 

1  Woepcke,  Recherches  sur  Vhistoire  des  sciences  matMmatiques  chez 
les  orientaux,  loc.  cit. ;  Propagation,  p.  57. 

2  Al-Hassar's  forms,  Suter,  Bibliotheca  Mathematica,  Vol.  II  (3), 
p.  15. 

3  "Woepcke,  Sur  une  donnie  historique,  etc.,  loc.  cit.  The  name  gobar 
is  not  used  in  the  text.  The  manuscript  from  which  these  are  taken 
is  the  oldest  (970  a.d.)  Arabic  document  known  to  contain  all  of  the 
numerals. 

4  Silvestre  de  Sacy,  loc.  cit.  He  gives  the  ordinary  modern  Arabic 
forms,  calling  them  Indien. 

5  and  6  Woepcke,  "Introduction  au  calcul  Gobari  et  Hawai,"  Atti 
delV  accademia  pontificia  dei  nuovi  Lincei,  Vol.  XIX.  The  adjective  ap- 
plied to  the  forms  in  5  is  gobari  and  to  those  in  6  indienne.  This  is  the 
direct  opposite  of  Woepcke's  use  of  these  adjectives  in  the  Recherches 
sur  Vhistoire  cited  above,  in  which  the  ordinary  Arabic  forms  (like 
those  in  row  5)  are  called  indiens. 

These  forms  are  usually  written  from  right  to  left. 


70  THE  HINDU-ARABIC  NUMERALS 

this  source  has  no  foundation.    The  first  four  Egyptian 
cardinal  numerals  1  resemble  more  the  modern  Arabic. 

1  I  This  theory  of  the  very  early 
_               introduction    of    the    numerals 

^  *■  into    Europe    fails    in    several 

2  j^     J&    points.    In  the  first  place  the 


^1    X  %  "J     early  "Western   forms   are   not 

'*  ^~»  *     known;    in    the    second  place 

<•••  2»i,  some  early  Eastern  forms  are 

2,^  a  L  hke  the  gobar,  as  is  seen  in  the 

*%  1  ^  ^  third  line  on  p.  69,  where  the 

-  ^T  «  forms  are   from   a   manuscript 

\t}  *V%  4****  written  at  Shiraz  about  970  A.D., 

\  *^^         and  in  which  some  western  Ara- 

/        /  /  hie  forms,  e.g.  |->  for  2,  are  also 

'     >c^  used.  Probably  most  significant 

Demotic  and  Hieratic      of  aR  ig  the  fact  that  the  •    Mr 

Ordinals  ,  ,      0 

numerals  as  given  by  bacy  are 

all,  with  the  exception  of  the  symbol  for  eight,  either  sin- 
gle Arabic  letters  or  combinations  of  letters.  So  much  for 
the  Woepcke  theory  and  the  meaning  of  the  gobar  numer- 
als. We  now  have  to  consider  the  question  as  to  whether 
Boethius  knew  these  gobar  forms,  or  forms  akin  to  them. 
This  large  question 2  suggests  several  minor  ones : 
(1)  Who  was  Boethius?  (2)  Could  he  have  known 
these  numerals?  (3)  Is  there  any  positive  or  strong  cir- 
cumstantial evidence  that  he  did  know  them  ?  (4)  What 
are  the  probabilities  in  the  case  ? 

1  J.  G.  Wilkinson,  The  Manners  and  Customs  of  the  Ancient  Egyp- 
tians, revised  by  S.  Birch,  London,  1878,  Vol.  II,  p.  493,  plate  XVI. 

2  There  is  an  extensive  literature  on  this  "  Boethius-Frage."  The 
reader  who  cares  to  go  fully  into  it  should  consult  the  various  volumes 
of  the  Juhrbuch  tiber  die  ForUschrilte  der  Mathematik. 


THE  BOETHIUS  QUESTION  71 

First,  who  was  Boethius,  —  Divus1  Boethius  as  he  was 
called  hi  the  Middle  Ages  ?  Anicius  Manlius  Severinus 
Boethius2  was  born  at  Rome  c.  475.  He  was  a  mem- 
ber of  the  distinguished  family  of  the  Anieii,3  which  had 
for  some  time  before  his  birth  been  Christian.  Early 
left  an  orphan,  the  tradition  is  that  he  was  taken  to 
Athens  at  about  the  age  of  ten,  and  that  he  remained 
there  eighteen  years.4  He  married  Rusticiana,  daughter 
of  the  senator  Symmachus,  and  this  union  of  two  such 
powerful  families  allowed  him  to  move  in  the  highest 
circles.5  Standing  strictly  for  the  right,  and  against  all 
iniquity  at  court,  he  became  the  object  of  hatred  on  the 
part  of  all  the  unscrupulous  element  near  the  throne, 
and  his  bold  defense  of  the  ex-consul  Albums,  unjustly 
accused  of  treason,  led  to  his  imprisonment  at  Pavia6 
and  his  execution  in  524.7  Not  many  generations  after 
his  death,  the  period  being  one  in  which  historical  criti- 
cism was  at  its  lowest  ebb,  the  church  found  it  profitable 
to  look  upon  his  execution  as  a  martyrdom.8    He  was 

1  This  title  was  first  applied  to  Roman  emperors  in  posthumous 
coins  of  Julius  Caesar.  Subsequently  the  emperors  assumed  it  during 
their  own  lifetimes,  thus  deifying  themselves.  See  F.  Gnecchi,  Monete 
romane,  2ded.,  Milan,  1000,  p.  200. 

2  This  is  the  common  spelling  of  the  name,  although  the  more  cor- 
rect Latin  form  is  Boetius.  See  Harper's  Dirt,  of  Class.  Lit.  and 
Antiq.,  New  York,  1807,  Vol.  I,  p.  213.  There  is  much  uncertainty  as 
to  his  life.  A  good  summary  of  the  evidence  is  given  in  the  last  two 
editions  of  the  Encyclopaedia  Britannica. 

3  His  father,  Flavius  Manlius  Boethius,  was  consul  in  487. 

4  There  is,  however,  no  good  historic  evidence  of  this  sojourn  in 
Athens. 

5  His  arithmetic  is  dedicated  to  Symmachus :  "  Domino  suo  patri- 
cio  Symmacho  Boetius.'1    [Friedlein  ed.,  p.  3.] 

6  It  was  while  here  that  he  wrote  De  consolatione  philosophiae. 

7  It  is  sometimes  given  as  525. 

8  There  was  a  medieval  tradition  that  he  was  executed  because  of  a 
work  ou  the  Trinity. 


72  THE  HINDU-ARABIC  NUMERALS 

accordingly  looked  upon  as  a  saint,1  his  bones  were  en- 
shrined,2 and  as  a  natural  consequence  his  books  were 
among  the  classics  in  the  church  schools  for  a  thousand 
years.3  It  is  pathetic,  however,  to  think  of  the  medieval 
student  trying  to  extract  mental  nourishment  from  a 
work  so  abstract,  so  meaningless,  so  unnecessarily  com- 
plicated, as  the  arithmetic  of  Boethius. 

He  was  looked  upon  by  his  contemporaries  and  imme- 
diate successors  as  a  master,  for  Cassiodorus4  (c.  490- 
c.  585  A.D.)  says  to  him :  "  Through  your  translations 
the  music  of  Pythagoras  and  the  astronomy  of  Ptolemy 
are  read  by  those  of  Italy,  and  the  arithmetic  of  Nicoma- 
chus  and  the  geometry  of  Euclid  are  known  to  those  of 
the  West."5     Founder   of   the   medieval  scholasticism, 

1  Hence  the  Divus  in  his  name. 

2  Thus  Dante,  speaking  of  his  burial  place  in  the  monastery  of  St. 
Pietro  in  Ciel  <T  Oro,  at  Pavia,  says : 

"  The  saintly  soul,  that  shows 
The  world's  deceitfulness,  to  all  who  hear  him, 
Is,  with  the  sight  of  all  the  good  that  is, 
Blest  there.   The  limbs,  whence  it  was  driven,  lie 
Down  in  Cieldauro ;  and  from  martyrdom 
And  exile  came  it  here."  —  Paradiso,  Canto  X. 

3  Not,  however,  in  the  mercantile  schools.  The  arithmetic  of  Boe- 
thius would  have  been  about  the  last  book  to  be  thought  of  in  such 
institutions.  While  referred  to  by  Bseda  (072-735)  and  Hrabanus 
Maurus  (c.  776-850),  it  was  only  after  Gerbert's  time  that  the  Bo'etii 
de  institutione  arithmetica  libri  duo  was  really  a  common  work. 

4  Also  spelled  Cassiodorius. 

5  As  a  matter  of  fact,  Boethius  could  not  have  translated  any  work 
by  Pythagoras  on  music,  because  there  was  no  such  work,  but  he  did 
make  the  theories  of  the  Pythagoreans  known.  Neither  did  he  trans- 
late Nicomachus,  although  he  embodied  many  of  the  ideas  of  the  Greek 
writer  in  his  own  arithmetic.  Gibbon  follows  Cassiodorus  in  these 
statements  in  his  Decline  and  Fall  of  the  Roman  Empire,  chap,  xxxix. 
Martin  pointed  out  with  positiveness  the  similarity  of  the  first  book 
of  Boethius  to  the  first  five  books  of  Nicomachus.  [Les  signcs  nume- 
ralix  etc.,  reprint,  p.  4.] 


THE  BOETHIUS  QUESTION  73 

distinguishing  the  trivium  and  quadrivium,1  writing  the 
only  classics  of  his  time,  Gibbon  well  called  him  "  the  last 
of  the  Romans  whom  Cato  or  Tully  could  have  acknowl- 
edged for  their  countryman."  2 

The  second  question  relating  to  Boethius  is  this :  Could 
he  possibly  have  known  the  Hindu  numerals  ?  In  view 
of  the  relations  that  will  be  shown  to  have  existed  be- 
tween the  East  and  the  West,  there  can  only  be  an 
affirmative  answer  to  this  question.  The  numerals  had 
existed,  without  the  zero,  for  several  centuries ;  they 
had  been  well  known  in  India  ;  there  had  been  a  contin- 
ued interchange  of  thought  between  the  East  and  West ; 
and  warriors,  ambassadors,  scholars,  and  the  restless  trader, 
all  had  gone  back  and  forth,  by  land  or  more  frequently 
by  sea,  between  the  Mediterranean  lands  and  the  centers 
of  Indian  commerce  and  culture.  Boethius  could  very 
well  have  learned  one  or  more  forms  of  Hindu  numerals 
from  some  traveler  or  merchant. 

To  justify  this  statement  it  is  necessary  to  speak  more 
fully  of  these  relations  between  the  Far  East  and  Europe. 
It  is  true  that  we  have  no  records  of  the  interchange  of 
learning,  in  any  large  way,  between  eastern  Asia  and 
central  Europe  in  the  century  preceding  the  time  of 
Boethius.  But  it  is  one  of  the  mistakes  of  scholars  to 
believe  that  they  are  the  sole  transmitters  of  knowledge. 

1  The  general  idea  goes  back  to  Pythagoras,  however. 

2  J.  C.  Scaliger  in  his  Poetice  also  said  of  him:  "Boethii  Severini 
ingenium,  eruditio,  ars,  sapientia  facile  provocat  omnes  auctores,  sive 
ill  I  Graeci  sint,  sive  Latini"  [Heilbronner,  Hist.  math,  univ.,  p.  387]. 
Libri,  speaking  of  the  time  of  Boethius,  remarks:  "Nous  voyons  du 
temps  de  Th^odoric,  les  lettres  reprendre  une  nouvelle  vie  en  Italie,  les 
6coles  florissantes  et  les  savans  honor&s.  Et  certes  les  ouvrages  de  Boece, 
de  Cassiodore,  de  Symmaque,  surpassent  de  beaucoup  toutes  les  pr<  >due- 
tions  du  siecle  pre^dent.1'    [Histoire  des  matliematiques,  Vol.  I,  p.  78.] 


74  THE  HINDU-ARABIC  NUMERALS 

As  a  matter  of  fact  there  is  abundant  reason  for  believ- 
ing that  Hindu  numerals  would  naturally  have  been 
known  to  the  Arabs,  and  even  along  every  trade  route 
to  the  remote  west,  long  before  the  zero  entered  to  make 
their  place-value  possible,  and  that  the  characters,  the 
methods  of  calculating,  the  improvements  that  took  place 
from  time  to  time,  the  zero  when  it  appeared,  and  the 
customs  as  to  solving  business  problems,  would  all  have 
been  made  known  from  generation  to  generation  along 
these  same  trade  routes  from  the  Orient  to  the  Occident. 
It  must  always  be  kept  in  mind  that  it  was  to  the  trades- 
man and  the  wandering  scholar  that  the  spread  of  such 
learning  was  due,  rather  than  to  the  school  man.  Indeed, 
Avicenna1  (980-1037  a.d.)  in  a  short  biography  of  him- 
self relates  that  when  his  people  were  living  at  Bokhara 
his  father  sent  him  to  the  house  of  a  grocer  to  learn  the 
Hindu  art  of  reckoning,  in  which  this  grocer  (oil  dealer, 
possibly)  was  expert.  Leonardo  of  Pisa,  too,  had  a  similar 
training. 

The  whole  question  of  this  spread  of  mercantile  knowl- 
edge along  the  trade  routes  is  so  connected  with  the  go- 
bar  numerals,  the  Boethius  question,  Gerbert,  Leonardo 
of  Pisa,  and  other  names  and  events,  that  a  digression 
for  its  consideration  now  becomes  necessary.2 

1  Carra  de  Vaux,  Avicenne,  Paris,  1000;  Woepcke,  Sur  Vintroduv- 
tion,  etc.;  Gerhardt,  Entstehung  etc.,  p.  20.  Avicenna  is  a  corruption 
from  Ibn  Rina,  as  pointed  out  by  Wiistenfeld,  Geschichte  der  arabischt  n 
Aerzte  und  Naturforscher,  Gottingen,  1840.  His  full  name  is  Abu  'All 
al-Hosein  ibn  Sina.  F.or  notes  on  Avicenna's  arithmetic,  see  Woepcke, 
Propagation,  p.  502. 

2  On  the  early  travel  between  the  East  and  the  West  the  follow- 
ing works  may  be  consulted:  A.  Hillebrandt,  Alt-Indien,  containing 
"Chinesisrliebeisendein  Indien,"  Breslau,  1899,  p.  179;  C.  A.  Rkeel, 
Travel  in  the  First  Century  after  Christ,  Cambridge,  1001,  p.  112;  M. 
Reinaud,  "  Relations  politiques  et  commerciales  de  I'empire  romain 


THE  BOETHIUS  QUESTION  75 

Even  in  very  remote  times,  before  the  Hindu  numer- 
als were  sculptured  in  the  cave  of  Nana  Ghat,  there  were 
trade  relations  between  Arabia  and  India.  Indeed,  long 
before  the  Aryans  went  to  India  the  great  Turanian  race 
had  spread  its  civilization  from  the  Mediterranean  to  the 
Indus.1  At  a  much  later  period  the  Arabs  were  the  inter- 
mediaries between  Egypt  and  Syria  on  the  west,  and  the 
farther  Orient.2  In  the  sixth  century  B.C.,  Hecatams,3 
the  father  of  geography,  was  acquainted  not  only  with  the 
Mediterranean  lands  but  with  the  countries  as  far  as  the 
Indus,4  and  in  Biblical  times  there  were  regular  triennial 
voyages  to  India.  Indeed,  the  story  of  Joseph  bears 
witness  to  the  caravan  trade  from  India,  across  Arabia, 
and  on  to  the  banks  of  the  Nile.  About  the  same  time 
as  Hecatams,  Scylax,  a  Persian  admiral  under  Darius, 
from  Caryanda  on  the  coast  of  Asia  Minor,  traveled  to 

avec  l'Asie  orientale,"  in  the  Journal  Asiatique,  Mars-Avril,  1863, 
Vol.  I  (0),  p.  93;  Beazley,  Dawn  of  Modern  Geography,  a  History  of 
Exploration  and  Geographical  Science  from  the  Conversion  of  the  Roman 
Empire  to  A.D.  1420,  London,  1897-1900,  3  vols.;  Heyd,  Geschichte  des 
Levanthandels  im  Mittelalter,  Stuttgart,  1897 ;  J.  Keane,  The  Evolution 
of  Geography,  London,  1899,  p.  38  ;  A.  Cunningham,  Corpus  inscriptio- 
num  Indicarum,  Calcutta,  1877,  Vol.  I ;  A.  Neander,  General  History 
of  the  Christian  Religion  and  Church,  5th  American  ed.,  Boston,  1855, 
Vol.  Ill,  p.  89 ;  R.  C.  Dutt,  A  History  of  Civilization  in  Ancient 
India,  Vol.  II,  Bk.  V,  chap,  ii ;  E.  C.  Bayley,  loc.  cit.,  p.  28  et  seq.; 
A.  C.  Burnell,  loc.  cit.,  p.  3 ;  J.  E.  Tennent,  Ceylon,  London,  1859, 
Vol.  I,  p.  159;  Geo.  Tumour,  Epitome  of  the  History  of  Ceylon,  Lon- 
don, n.d.,  preface;  " Philalethes,"  History  of  Ceylon,  London,  1810, 
chap,  i;  H.  C.  Sirr,  Ceylon  and  the  Cingalese,  London,  1850,  Vol.  I, 
chap.  ix.  On  the  Hindu  knowledge  of  the  Nile  see  E.  Wilford,  Asi- 
atick  Researches,  Vol.  Ill,  p.  295,  Calcutta,  1792. 

1  G.  Oppert,  On  the  Ancient  Commerce  of  India,  Madras,  1879,  p.  8. 

2  Gerhardt,  Etudes  etc.,  pp.  8,  11. 

3  See  Smith's  Dictionary  of  Greek  and  Roman  Biography  and  Mythol- 
ogy. 

4  P.  M.  Sykes,  Ten  Thousand  Miles  in  Persia,  or  Eight  Years  in 
Iran,  London,  1902,  p.  107.  Sykes  was  the  first  European  to  follow 
the  course  of  Alexander's  army  across  eastern  Persia. 


76  THE  HINDU-ARABIC  NUMERALS 

northwest  India  and  wrote  upon  his  ventures.1  He  induced 
the  nations  along  the  Indus  to  acknowledge  the  Persian 
supremacy,  and  such  number  systems  as  there  were  in 
these  lands  would  naturally  have  been  known  to  a  man 
of  his  attainments. 

A  century  after  Scylax,  Herodotus  showed  consider- 
able knowledge  of  India,  speaking  of  its  cotton  and  its 
gold,2  telling  how  Sesostris 3  fitted  out  ships  to  sail  to 
that  country,  and  mentioning  the  routes  to  the  east. 
These  routes  were  generally  by  the  Red  Sea,  and  had 
been  followed  by  the  Phoenicians  and  the  Sabasans,  and 
later  were  taken  by  the  Greeks  and  Romans.4 

In  the  fourth  century  B.C.  the  West  and  East  came  into 
very  close  relations.  As  early  as  330,  Pytheas  of  Mas- 
silia  (Marseilles)  had  explored  as  far  north  as  the  north- 
ern end  of  the  British  Isles  and  the  coasts  of  the  German 
Sea,  while  Macedon,  in  close  touch  with  southern  France, 
was  also  sending  her  armies  under  Alexander  5  through 
Afghanistan  as  far  east  as  the  Punjab.6  Pliny  tells  us 
that  Alexander  the  Great  employed  surveyors  to  measure 

1  Biihler,  Indian  Brahma  Alphabet,  note,  p.  27  ;  Palaeographie,  p.  2  ; 
Uerodoti  Halicarnassei  Mstoria,  Amsterdam,  1763,  Bk.  IV,  p.  300; 
Isaac  Vossius,  Periplus  Scylacis  Caryandensis,  1039.  It  is  doubtful 
whether  the  work  attributed  to  Scylax  was  written  by  him,  but  in 
any  case  the  work  dates  back  to  the  fourth  century  b.c.  See  Smith's 
Dictionary  of  Greek  and  Roman  Biography. 

2  Herodotus,  Bk.  III. 

3  RainesesII(?),  the  Sesoosis  of  Diodorus  Siculus. 

4  Indian  Antiquary,  Vol.  I,  p.  229;  F.  B.  Jevons,  Manual  of  Greek 
A  nMquities,  London,  1895,  p.  386.  On  the  relations,  political  and  com- 
mercial, between  India  and  E.nypt  c.  72  B.C.,  under  Ptolemy  Auletes, 
see  the  Journal  Asinliipic,  1863,  p.  297. 

5  Sikamlar,  as  the  name  still  remains  in  northern  India. 

e  Harper's  Classical  Diet.,  New  York,  1897,  Vol.  I,  p.  724;  F.  B. 
Jevons,  loc.  cit.,  p.  389;  J.  C.  Marslnnan,  Abridyiuud  of  the  JHator-y 
of  India,  chaps,  i  and  ii. 


THE  BOETHIUS  QUESTION  77 

the  roads  of  India;  and  one  of  the  great  highways  is 
described  by  Megasthenes,  who  in  295  B.C.,  as  the  ambas- 
sador of  Seleucus,  resided  at  Pataliputra,  the  present 
Patna.1 

The  Hindus  also  learned  the  art  of  coining  from  the 
Greeks,  or  possibly  from  the  Chinese,  and  the  stores  of 
Greco-Hindu  coins  still  found  in  northern  India  are  a 
constant  source  of  historical  information.2  The  Rama- 
yana  speaks  of  merchants  traveling  in  great  caravans 
and  embarking  by  sea  for  foreign  lands.3  Ceylon  traded 
with  Malacca  and  Siam,  and  Java  was  colonized  by  Hindu 
traders,  so  that  mercantile  knowledge  was  being  spread 
about  the  Indies  during  all  the  formative  period  of  the 
numerals. 

Moreover  the  results  of  the  early  Greek  invasion  were 
embodied  by  Dicsearchus  of  Messana  (about  320  B.C.)  in 
a  map  that  long  remained  a  standard.  Furthermore, 
Alexander  did  not  allow  his  influence  on  the  East  to 
cease.  He  divided  India  into  three  satrapies,4  placing 
Greek  governors  over  two  of  them  and  leaving  a  Hindu 
ruler  in  charge  of  the  third,  and  in  Bactriana,  a  part  of 
Ariana  or  ancient-  Persia,  he  left  governors  ;  and  hi  these 
the  western  civilization  was  long  in  evidence.  Some  of 
the  Greek  and  Roman  metrical  and  astronomical  terms 

1  Oppert,  loc.  cit.,  p.  11.  It  was  at  or  near  this  place  that  the  first 
great  Indian  mathematician,  Aryabhata,  was  born  in  476  a.d. 

2  Biihler,  Palaeographie,  p.  2,  speaks  of  Greek  coins  of  a  period 
anterior  to  Alexander,  found  in  northern  India.  More  complete  infor- 
mation may  be  found  in  Indian  Coins,  by  E.  J.  Rapson,  Strassburg, 
1898,  pp.  3-7. 

3  Oppert,  loc.  cit.,  p.  14 ;  and  to  him  is  due  other  similar  infor- 
mation. 

4  J.  Beloch,  Griechische  GeschicMe,  Vol.  Ill,  Strassburg,  1904,  pp. 
30-31. 


78  THE  HINDU-ARABIC  NUMERALS 

found  their  way,  doubtless  at  this  time,  into  the  Sanskrit 
language.1  Even  as  late  as  from  the  second  to  the  fifth 
centuries  A.D.,  Indian  coins  showed  the  Hellenic  influ- 
ence. The  Hindu  astronomical  terminology  reveals  the 
same  relationship  to  western  thought,  for  Varaha-Mihira 
(6th  century  a.d.),  a  contemporary  of  Aryabhata,  enti- 
tled a  work  of  his  the  Brhat-Savihitd,  a  literal  translation 
of  fiejaXi]  avvTct^is  of  Ptolemy;2  and  in  various  ways  is 
this  interchange  of  ideas  apparent.3  It  could  not  have 
been  at  all  unusual  for  the  ancient  Greeks  to  go  to  In- 
dia, for  Strabo  lays  down  the  route,  saying  that  all  who 
make  the  journey  start  from  Ephesus  and  traverse  Phrygia 
and  Cappadocia  before  taking  the  direct  road.4  The  prod- 
ucts of  the  East  were  always  finding  their  way  to  the 
West,  the  Greeks  getting  their  ginger5  from  Malabar, 
as  the  Phoenicians  had  long  before  brought  gold  from 
Malacca. 

Greece  must  also  have  had  early  relations  with  China, 
for  there  is  a  notable  similarity  between  the  Greek  and 
Chinese  life,  as  is  shown  in  their  houses,  their  domestic 
customs,  their  marriage  ceremonies,  the  public  story- 
tellers, the  puppet  shows  which  Herodotus  says  were 
introduced  from  Egypt,  the  street  jugglers,  the  games  of 
dice,6  the  game  of  finger-guessing,7  the  water  clock,  the 

1  E.g.,  the  denarius,  the  words  for  hour  and  minute  (upa,  \ewr6v), 
and  possibly  the  signs  of  the  zodiac.  [R.  Caldwell,  Comparative  Gram- 
mar of  the  Dravidian  Languages,  London,  1856,  p.  438.]  On  the  prob- 
able Chinese  origin  of  the  zodiac  see  Schlegel,  loc.  cit. 

2  Marie,  Vol.  II,  p.  73 ;  R.  Caldwell,  loc.  cit. 

3  A.  Cunningham,  loc.  cit.,  p.  50. 

4  C.  A.  J.  Skeel,  Travel,  loc.  cit.,  p.  14. 

5  Inchiver,  from  inchi,  "the  green  root."  [Indian  Antiquary,  Vol.  I, 
p.  352.] 

6  In  China  dating  only  from  the  second  century  a.d.,  however. 

7  The  Italian  viorra. 


THE  BOETHIUS  QUESTION  79 

music  system,  the  use  of  the  myriad,1  the  calendars,  and 
in  many  other  ways.2  In  passing  through  the  suburbs  of 
Peking  to-day,  on  the  way  to  the  Great  Bell  temple,  one 
is  constantly  reminded  of  the  semi-Greek  architecture  of 
Pompeii,  so  closely  does  modern  China  touch  the  old 
classical  civilization  of  the  Mediterranean.  The  Chinese 
historians  tell  us  that  about  200  B.C.  their  arms  were  suc- 
cessful in  the  far  west,  and  that  in  180  B.C.  an  ambassador 
went  to  Bactria,  then  a  Greek  city,  and  reported  that  Chi- 
nese products  were  on  sale  in  the  markets  there.3  There 
is  also  a  noteworthy  resemblance  between  certain  Greek 
and  Chinese  words,4  showing  that  in  remote  times  there 
must  have  been  more  or  less  interchange  of  thought. 

The  Romans  also  exchanged  products  with  the  East. 
Horace  says,  "  A  busy  trader,  you  hasten  to  the  farthest 
Indies,  flying  from  poverty  over  sea,  over  crags,  over 
fires."  5  The  products  of  the  Orient,  spices  and  jewels 
from  India,  frankincense  from  Persia,  and  silks  from 
China,  being  more  in  demand  than  the  exports  from  the 
Mediterranean  lands,  the  balance  of  trade  was  against 
the  West,  and  thus  Roman  coin  found  its  way  east- 
ward. In  1898,  for  example,  a  number  of  Roman  coins 
dating  from  111  B.C.  to  Hadrian's  time  were  found  at 
Pakli,  a  part  of  the  Hazara  district,  sixteen  miles  north 
of  Abbottabad,6  and  numerous  similar  discoveries  have 
been  made  from  time  to  time. 

1  J.  Bowring,  The  Decimal  System,  London,  1854,  p.  2. 

2  II.  A.  Giles,  lecture  at  Columbia  University,  March  12,  1902,  on 
"China  and  Ancient  Greece."  3  Giles,  loc.  cit. 

4  E.g.,  the  names  for  grape,  radish  (la-po,  pdcpTj),  water-lily  (si-kua, 
"west  gourds";  criKifo,  "gourds"),  are  much  alike.    [Giles,  loc.  cit.] 

5  Epistles,  I,  1,  45-40.  On  the  Roman  trade  routes,  see  Beazley, 
loc.  cit.,  Vol.  I,  p.  170. 

6  Am.  Journ.  of  Archeol.,  Vol.  IV,  p.  360. 


80  THE  HINDU-ARABIC  NUMERALS 

Augustus  speaks  of  envoys  received  by  him  from  India, 
a  thing  never  before  known,1  and  it  is  not  improbable  that 
he  also  received  an  embassy  from  China.2  Suetonius  (first 
century  a.d.)  speaks  in  his  history  of  these  relations,3  as 
do  several  of  his  contemporaries,4  and  Vergil 5  tells  of 
Augustus  doing  battle  in  Persia.  In  Pliny's  time  the 
trade  of  the  Roman  Empire  with  Asia  amounted  to  a 
million  and  a  quarter  dollars  a  year,  a  sum  far  greater 
relatively  then  than  now,6  while  by  the  time  of  Constan- 
tine  Europe  was  in  direct  communication  with  the  Far 
East.7 

In  view  of  these  relations  it  is  not  beyond  the  range  of 
possibility  that  proof  may  sometime  come  to  light  to  show 
that  the  Greeks  and  Romans  knew  something  of  the 

1  M.  Perrot  gives  this  conjectural  restoration  of  his  words:  "Ad 
me  ex  India  regum  legationes  saepe  missi  sunt  numquam  antea  visae 
apud  quemquain  principem  Romanorum."  [M.  Reinaud,  "Relations 
politiques  et  commerciales  de  1'empire  romain  avec  l'Asie  orientale," 
Journ.  Asiat.,  Vol.  I  (6),  p.  93.] 

2  Reinaud,  loc.  cit.,  p.  189.  Floras,  II,  34  (IV,  12),  refers  to  it: 
"  Seres  etiam  habitantesque  sub  ipso  sole  Indi,  cum  gemmis  et  margari- 
tis  elephantes  quoque  inter  munera  trahentes  nihil  magis  quam  longin- 
quitatem  viae  imputabant."  Horace  shows  his  geographical  knowledge 
by  saying :  "  Not  those  who  drink  of  the  deep  Danube  shall  now  break 
the  Julian  edicts;  not  the  Getae,  not  the  Seres,  nor  the  perfidious 
Persians,  nor  those  Jborn  on  the  river  Tanai's."  [Odes,  Bk.  IV,  Ode 
15,  21-24.] 

3  "  Qua virtutis  moderationisque  fama  Indos  etiam  ac  Scythasauditu 
modo  cognitos  pellexit  ad  amicitiam  suam  populique  Romani  ultro  per 
legatos  petendam.,1    [Reinaud,  loc.  cit.,  p.  180.] 

4  Reinaud,  loc.  cit.,  p.  180. 

5  Georgits,  II,  170-172.    So  Propertius  (Elegies,  III,  4): 

Anna  deus  Caesar  dites  meditatur  ad  Indos 
Et  freta  gemmiferi  flndere  classe  maris. 

"Tlic  divine  Caesar  meditated  carrying  arms  against  opulent  India,  and 
witli  his  ships  to  cut  the  gem-bearing  seas." 

6  Heyd,  loc.  cit.,  Vol.  I,  p.  4. 

7  Reinaud,  loc.  cit.,  p.  393. 


THE  BOETHIUS  QUESTION  81 

number  system  of  India,  as  several  writers  have  main- 
tained.1 

Returning  to  the  East,  there  are  many  evidences  of  the 
spread  of  knowledge  in  and  about  India  itself.  In  the 
third  century  B.C.  Buddhism  began  to  be  a  connecting 
medium  of  thought.  It  had  already  permeated  the  Hima- 
laya territory,  had  reached  eastern  Turkestan,  and  had 
probably  gone  thence  to  China.  Some  centuries  later  (in 
62  a.d.)  the  Chinese  emperor  sent  an  ambassador  to 
India,  and  in  67  a.d.  a  Buddhist  monk  was  invited  to 
China.2  Then,  too,  in  India  itself  Asoka,  whose  name 
has  already  been  mentioned  in  this  work,  extended  the 
boundaries  of  his  domains  even  into  Afghanistan,  so  that 
it  was  entirely  possible  for  the  numerals  of  the  Punjab 
to  have  worked  their  way  north  even  at  that  early  date.3 

Furthermore,  the  influence  of  Persia  must  not  be  for- 
gotten in  considering  this  transmission  of  knowledge.  In 
the  fifth  century  the  Persian  medical  school  at  Jondi- 
Sapur  admitted  both  the  Hindu  and  the  Greek  doctrines, 
and  Firdusi  tells  us  that  during  the  brilliant  reign  of 

1  The  title  page  of  Calanclri  (1491),  for  example,  represents  Pythago- 
ras with  these  numerals  before  him.  [Smith,  Rara  Arithmetica,  p.  46.] 
Isaacus  Vossius,  Observationes  ad  Pomponium  Melam  de  situ  orbis,  1658, 
maintained  that  the  Arabs  derived  these  numerals  from  the  west.  A 
learned  dissertation  to  this  effect,  but  deriving  them  from  the  Romans 
instead  of  the  Greeks,  was  written  by  Ginanni  in  1753  (Dissertatio 
mathematica  critica  de  numeralium  notarum  minuscularum  origine,  Ven- 
ice, 1753).  See  also  Mannert,  De  numerorum  quos  arabicos  vocant  vera 
origine  Pythagorica,  Nurnberg,  1801.  Even  as  late  as  1827  Romagnosi 
(in  his  supplement  to  Ricerche  storiche  sulV  India  etc.,  by  Robertson, 
Vol.  II,  p.  580,  1827)  asserted  that  Pythagoras  originated  them.  [R. 
Bombelli,  Vantica  numerazione  italica,  Rome,  1876,  p.  59.]  Gow  (Hist, 
of  Greek  Math.,  p.  98)  thinks  that  Iamblichus  must  have  known  a  simi- 
lar system  in  order  to  have  worked  out  certain  of  his  theorems,  but 
this  is  an  unwarranted  deduction  from  the  passage  given. 

2  A.  Hillebrandt,  Alt-Indien,  p.  179. 

3  J.  C.  Marshman,  loc.  cit.,  chaps,  i  and  ii. 


82  THE  HINDU-ARABIC  NUMERALS 

Khosru  I,1  the  golden  age  of  Pahlavi  literature,  the 
Hindu  game  of  chess  was  introduced  into  Persia,  at  a 
time  when  wars  with  the  Greeks  were  bringing  prestige 
to  the  Sassanid  dynasty. 

Again,  not  far  from  the  time  of  Boethius,  in  the  sixth 
century,  the  Egyptian  monk  Cosmas,  in  his  earlier  years 
as  a  trader,  made  journeys  to  Abyssinia  and  even  to 
India  and  Ceylon,  receiving  the  name  Indicoplmstes  (the 
Indian  traveler).  His  map  (547  A. d.)  shows  some  knowl- 
edge of  the  earth  from  the  Atlantic  to  India.  Such  a 
man  would,  with  hardly  a  doubt,  have  observed  every 
numeral  system  used  by  the  people  with  whom  he  so- 
journed,2 and  whether  or  not  he  recorded  his  studies  in 
permanent  form  he  would  have  transmitted  such  scraps 
of  knowledge  by  word  of  mouth. 

As  to  the  Arabs,  it  is  a  mistake  to  feel  that  their  activi- 
ties began  with  Mohammed.  Commerce  had  always  been 
held  in  honor  by  them,  and  the  Qoreish  3  had  annually 
for  many  generations  sent  caravans  bearing  the  spices  and 
textiles  of  Yemen  to  the  shores  of  the  Mediterranean.  In 
the  fifth  century  they  traded  by  sea  with  India  and  even 
with  China,  and  Hira  was  an  emporium  for  the  wares  of 
the  East,4  so  that  any  numeral  system  of  any  part  of  the 
trading  world  could  hardly  have  remained  isolated. 

Long  before  the  warlike  activity  of  the  Arabs,  Alex- 
andria had  become  the  great  market-place  of  the  world. 
From  this  center  caravans  traversed  Arabia  to  Hadra- 
niaut,  where  they  met  ships  from  India.  Others  went 
north  to  Damascus,  while  still  others  made  their  way 

1  Tie  reigned  531-579  a.d.;  called  Nusirwan,  the  holy  one. 

2  J.  Kcanc,  The  Evolution  of  Geography,  London,  1899,  p.  38. 

3  The  Arabs  who  lived  in  and  about  Mecca. 

4  S.  Guyard,  in  Encye.  Brit.,  9th  ed.,  Vol.  XVI,  p.  597. 


THE  BOETIIIUS  QUESTION  83 

along  the  southern  shores  of  the  Mediterranean.  Ships 
sailed  from  the  isthmus  of  Suez  to  all  the  commercial 
ports  of  Southern  Europe  and  up  into  the  Black  Sea. 
Hindus  were  found  among  the  merchants 1  who  fre- 
quented the  bazaars  of  Alexandria,  and  Brahmins  were 
reported  even  in  Byzantium. 

Such  is  a  very  brief  resume  of  the  evidence  showing 
that  the  numerals  of  the  Punjab  and  of  other  parts  of 
India  as  well,  and  indeed  those  of  China  and  farther 
Persia,  of  Ceylon  and  the  Malay  peninsula,  might  well 
have  been  known  to  the  merchants  of  Alexandria,  and 
even  to  those  of  any  other  seaport  of  the  Mediterranean, 
in  the  time  of  Boethius.  The  Bralmii  numerals  would 
not  have  attracted  the  attention  of  scholars,  for  they  had 
no  zero  so  far  as  we  know,  and  therefore  they  were  no 
better  and  no  worse  than  those  of  dozens  of  other  sys- 
tems. If  Boethius  was  attracted  to  them  it  was  probably 
exactly  as  any  one  is  naturally  attracted  to  the  bizarre 
or  the  mystic,  and  he  would  have  mentioned  them  in  his 
works  only  incidentally,  as  indeed  they  are  mentioned  in 
the  manuscripts  in  which  they  occur. 

In  answer  therefore  to  the  second  question,  Could 
Boethius  have  known  the  Hindu  numerals  ?  the  reply 
must  be,  without  the  slightest  doubt,  that  he  could  easily 
have  known  them,  and  that  it  would  have  been  strange 
if  a  man  of  his  inquiring  mind  did  not  pick  up  many 
curious  bits  of  information  of  this  kind  even  though  he 
never  thought  of  making  use  of  them. 

Let  us  now  consider  the  third  question,  Is  there  any 
positive  or  strong  circumstantial  evidence  that  Boethius 
did  know  these  numerals  ?    The  question  is  not  new, 

1  Oppert,  loc.  cit.,  p.  29. 


84  THE  HINDU-ARABIC  NUMERALS 

nor  is  it  much  nearer  being  answered  than  it  was  over 
two  centuries  ago  when  Wallis  (1693)  expressed  his 
doubts  about  it1  soon  after  Vossius  (1658)  had  called 
attention  to  the  matter.2  Stated  briefly,  there  are  three 
works  on  mathematics  attributed  to  Boethius : 3  (1)  the 
arithmetic,  (2)  a  work  on  music,  and  (3)  the  geometry.4 
The  genuineness  of  the  arithmetic  and  the  treatise  on 
music  is  generally  recognized,  but  the  geometry,  which 
contains  the  Hindu  numerals  with  the  zero,  is  under 
suspicion.5  There  are  plenty  of  supporters  of  the  idea 
that  Boethius  knew  the  numerals  and  included  them  in 
this  book,6  and  on  the  other  hand  there  are  as  many  who 

1  "At  non  credenduni  est  id  in  Autographis  contigisse,  aut  vetusti- 
oribus  Codd.  MSS."    [Wallis,  Opera  omnia,  Vol.  II,  p.  11.] 

2  In  Observationes  ad  Pomponium  Melam  de  situ  orbis.  The  ques- 
tion was  next  taken  up  in  a  large  way  by  Weidler,  loc.  cit.,  De  charac- 
teribus  etc.,  1727,  and  in  Spicilegium  etc.,  1755. 

3  The  best  edition  of  these  works  is  that  of  G.  Friedlein,  Anicii 
Manlii  Torquati  Severini  Boetii  de  institutione  arithmetical  libri  duo,  de 
institutione  musica  libri  quinque.  Accedit  geometria  quae  fertur  Boetii. 
.  .  .  Leipzig.  .  .  .  mdccclxvii. 

4  See  also  P.  Tannery,  "  Notes  sur  la  pseudo-g6om6trie  de  Boece," 
in  Bibliotheca  Mathematica,  Vol.  I  (3),  p.  39.  This  is  not  the  geometry 
in  two  books  in  which  are  mentioned  the  numerals.  There  is  a  manu- 
script of  this  pseudo-geometry  of  the  ninth  century,  but  the  earliest 
one  of  the  other  work  is  of  the  eleventh  century  (Tannery),  unless 
the  Vatican  codex  is  of  the  tenth  century  as  Friedlein  (p.  372)  asserts. 

5  Friedlein  feels  that  it  is  partly  spurious,  but  he  says:  " Eorum 
librorum,  quos  Boetius  de  geometria  scripsisse  dicitur,  investigare 
veram  inscriptionem  nihil  aliud  esset  nisi  operam  et  tempus  perdere." 
[Preface,  p.  v.]  N.  Bubnov  in  the  Russian  Journal  of  the  Ministry  of 
Public  Instruction,  1007,  in  an  article  of  which  a  synopsis  is  given  in 
the  Jahrbuch  iiber  die  Fortschritte  der  Mathematik  for  1007,  asserts  that 
the  geometry  was  written  in  the  eleventh  century. 

6  The  most  noteworthy  of  these  was  for  a  long  time  Cantor  (Ge- 
schichte,  Vol.  L,  3d  ed.,  pp.  587-588),  who  in  his  earlier  days  even 
believed  that  Pythagoras  had  known  them.  Cantor  says  (Die  romischen 
Agrimensoren,  Leipzig,  1875,  p.  130):  "  Uns  also,  wir  wiederholen  es, 
ist  die  Geometrie  des  Boetius  echt,  dieselbe  Schrift,  welche  er  nach 
Euklid  bearbeitete,  von  welcher  ein  Codex  bereits  in  Jahre  821  im 


THE  BOETHIUS  QUESTION  85 

feel  that  the  geometry,  or  at  least  the  part  mentioning 
the  numerals,  is  spurious.1  The  argument  of  those  who 
deny  the  authenticity  of  the  particular  passage  in  ques- 
tion may  briefly  be  stated  thus  : 

1.  The  falsification  of  texts  has  always  been  the  sub- 
ject of  complaint.  It  was  so  with  the  Romans,2  it  was  com- 
mon in  the  Middle  Ages,3  and  it  is  much  more  prevalent 

Kloster  Reichenau  vorhanden  war,  von  welcher  ein  anderes  Exemplar 
im  Jahre  982  zu  Mantua  in  die  Hande  Gerbert's  gelangte,  von  welcher 
mannigfache  Handschriften  noch  heute  vorhanden  sind."  But  against 
this  opinion  of  the  antiquity  of  MSS.  containing  these  numerals  is 
the  important  statement  of  P.  Tannery,  perhaps  the  most  critical  of 
modern  historians  of  mathematics,  that  none  exists  earlier  than  the 
eleventh  century.  See  also  J.  L.  Heiberg  in  Philologus,  Zeitschrift  f. 
d.  Mass.  Altertum,  Vol.  XLIII,  p.  508. 

<  >f  Cantor's  predecessors,  Th.  H.  Martin  was  one  of  the  most  promi- 
nent, his  argument  for  authenticity  appearing  in  the  Revue  ArcMolo- 
gique  for  1856-1857,  and  in  his  treatise  Les  signes  num&raux  etc. 
See  also  M.  Chasles,  "De  la  connaissance  qu'ont  eu  les  anciens  d'une 
numeration  derimale  e"crite  qui  fait  usage  de  neuf  chiffres  prenant 
les  valeurs  de  position,"  Comptes  rendus,  Vol.  VI,  pp.  678-680;  "Sur 
l'origine  de  notre  systeme  de  numeration,"  Comptes  rendus,  Vol. 
VIII,  pp.  72-81 ;  and  note  "Sur  le  passage  du  premier  livre  de  la  geo- 
metric de  Boece,  relatif  a  un  nouveau  systeme  de  numeration,"  in  his 
work  Apert'u  historique  sur  Vorigine  et  le  developpement  des  methodes  en 
geomelrie,  of  which  the  first  edition  appeared  in  1837. 

1  J.  L.  Heiberg  places  the  book  in  the  eleventh  century  on  philo- 
logical grounds,  Philologus,  loc.  cit. ;  Woepcke,  in  Propagation,  p.  44 ; 
Blume,  Lachmann,  and  Rudorff,  Die  Schriften  der  romischen  Feldmesser, 
Berlin,  1848 ;  Boeckh,  De  abaco  graecorum,  Berlin,  1841 ;  Friedlein, 
in  his  Leipzig  edition  of  1867 ;  Weissenborn,  Abhandlungen,  Vol.  II, 
p.  185,  his  Gerbert,  pp.  1,  247,  and  his  Geschichte  der  Einfiihrung  der 
jetzigen  Ziffern  in  Europa  durch  Gerbert,  Berlin,  1892,  p.  11 ;  Bayley, 
loc.  cit.,  p.  59;  Gerhardt,  Etudes,  p.  17,  Entstehung  und  Ausbreitung, 
p.  14 ;  Nagl,  Gerbert,  p.  57 ;  Bubnov,  loc.  cit.  See  also  the  discussion 
by  Chasles,  Halliwell,  and  Libri,  in  the  Comptes  rendus,  1839,  Vol.  IX, 
p.  447,  and  in  Vols.  VIII,  XVI,  XVII  of  the  same  journal. 

2  J.  Marquardt,  La  vie  privee  des  Romains,  Vol.  II  (French  trans.), 
p.  505,  Paris,  1893. 

3  In  a  Plimpton  manuscript  of  the  arithmetic  of  Boethius  of  the  thir- 
teenth century,  for  example,  the  Roman  numerals  are  all  replaced  by 
the  Arabic,  and  the  same  is  true  in  the  first  printed  edition  of  the  book. 


86  THE  HINDU-ARABIC  NUMERALS 

to-day  than  we  commonly  think.  We  have  but  to  see 
how  every  hymn-book  compiler  feels  himself  author- 
ized to  change  at  will  the  classics  of  our  language,  and 
how  unknown  editors  have  mutilated  Shakespeare,  to  see 
how  much  more  easy  it  was  for  medieval  scribes  to  insert 
or  eliminate  paragraphs  without  any  protest  from  critics.1 

2.  If  Boethius  had  known  these  numerals  he  would  have 
mentioned  them  in  his  arithmetic,  but  he  does  not  do  so.2 

3.  If  he  had  known  them,  and  had  mentioned  them  in 
any  of  his  works,  his  contemporaries,  disciples,  and  suc- 
cessors would  have  known  and  mentioned  them.  But 
neither  Capella  (c.  475)3  nor  any  of  the  numerous  medi- 
eval writers  who  knew  the  works  of  Boethius  makes  any 
reference  to  the  system.4 

(See  Smith's  Eara  Arithmetical  pp.434,  25-27.)  D.  E.  Smith  also  cop- 
ied from  a  manuscript  of  the  arithmetic  in  the  Laurentian  library  at 

Florence,  of  1370,  the  following  forms,  /  ^2  ~^  ol  ^  C  ~\  ^  1  ° 
which,  of  course,  are  interpolations.  An  interesting  example  of  a  for- 
gery in  ecclesiastical  matters  is  in  the  charter  said  to  have  been  given 
by  St.  Patrick,  granting  indulgences  to  the  benefactors  of  Glastonbury, 
dated  "In  nomine  domini  nostri  Jhesu  Christi  Ego  Patricius  humilis 
servunculus  Dei  anno  incarnationis  ejusdem  ccccxxx."  Now  if  the 
Benedictines  are  right  in  saying  that  Dionysius  Exiguus,  a  Scythian 
monk,  first  arranged  the  Christian  chronology  c.  532  a.d.,  this  can 
hardly  be  other  than  spurious.    See  Arbuthnot,  loc.  cit.,  p.  38. 

1  Halliwell,  in  his  Eara  Mathematica,  p.  107,  states  that  the  disputed 
passage  is  not  in  a  manuscript  belonging  to  Mr.  Ames,  nor  in  one  at 
Trinity  College.  See  also  Woepcke,  in  Propagation,  pp.  37  and  42. 
It  was  the  evident  corruption  of  the  texts  in  such  editions  of  Boethius 
as  those  of  Venice,  1490,  Easel,  1546  and  1570,  that  led  Woepcke 
to  publish  his  work  Sur  V  introduction  de  V arithrne'tique  indienne  en 
Occident. 

2  They  are  found  in  none  of  the  very  ancient  manuscripts,  as,  for 
example,  in  the  ninth-century  (?)  codex  in  the  Laurentian  library 
which  one  of  the  authors  has  examined.  It  should  be  said,  however, 
that  the  disputed  passage  was  written  after  the  arithmetic,  for  it  con- 
tains a  reference  to  that  work.    See  the  Friedlein  ed.,  p.  397. 

8  Smith,  Eara  Arithmetica,  p.  66. 

4  J.  L.  Ileiberg,  Philologus,  Vol.  XLIII,  p.  507. 


THE  BOETHIUS  QUESTION  87 

4.  The  passage  in  question  has  all  the  appearance  of 
an  interpolation  by  some  scribe.  Boethius  is  speaking  of 
angles,  in  his  work  on  geometry,  when  the  text  suddenly 
changes  to  a  discussion  of  classes  of  numbers.1  This  is 
followed  by  a  chapter  in  explanation  of  the  abacus,2  in 
which  are  described  those  numeral  forms  which  are  called 
apices  or  caracteres.3  The  forms  4  of  these  characters  vary 
in  different  manuscripts,  but  in  general  are  about  as 
shown  on  page  88.  They  are  commonly  written  with 
the  9  at  the  left,  decreasing  to  the  unit  at  the  right,  nu- 
merous writers  stating  that  this  was  because  they  were 
derived  from  Semitic  sources  in  which  the  direction  of 
writing  is  the  opposite  of  our  own.  This  practice  con- 
tinued until  the  sixteenth  century.5  The  writer  then 
leaves  the  subject  entirely,  using  the  Roman  numerals 

1  "Nosse  autem  huius  artis  clispicientem,  quid  sint  digiti,  quid  arti- 
culi,  quid  compositi,  quid  incompositi  numeri."   [Friedlein  ed.,  p. 395.] 

2  I)e  ratione  abaci.  In  this  he  describes  "  quandam  formulam,  quam 
ob  honorem  sui  praeceptoris  mensam  Pythagoream  nominabant  .  .  . 
a  posterioribus  appellabatur  abacus."  This,  as  pictured  in  the  text,  is 
the  common  Gerbert  abacus.  In  the  edition  in  Migne's  Patrologia 
Latina,  Vol.  LXIII,  an  ordinary  multiplication  table  (sometimes  called 
Pythagorean  abacus)  is  given  in  the  illustration. 

3  "  Habebant  enim  diverse  f  ormatos  apices  vel  caracteres."  See  the 
reference  to  Gerbert  on  p.  117. 

4  C.  Henry,  "  Sur  l'origine  de  quelques  notations  math&natiques," 
Revue  Archeologique,  1879,  derives  these  from  the  initial  letters  used  as 
abbreviations  for  the  names  of  the  numerals,  a  theory  that  finds  few 
supporters. 

5  E.g.,  it  appears  in  Schonerus,  Algorithmus  Demonstrates,  Niirn- 
berg,  1534,  f.  A  4.  In  England  it  appeared  in  the  earliest  English 
arithmetical  manuscript  known,  The  Crafte  of  Nombrynge :  "Ifforther- 
more  ye  most  vndirstonde  that  in  this  craft  ben  vsid  teen  figurys,  as 
here  bene  writen  for  ensampul,  0  9  8  A  6  4  <?  3  2  1  .  .  .  in  the  quych  we 
vse  teen  figurys  of  Inde.  Questio.  If  why  ten  f  yguris  of  Inde  ?  Solu- 
cio.  for  as  I  have  sayd  afore  thei  were  f onde  fyrst  in  Inde  of  a  kynge 
of  that  Cuntre,  that  was  called  Algor."  See  Smith,  An  Early  English 
Algorism,  loc.  cit. 


88  THE  HINDU-ARABIC  NUMERALS 

Forms  of  the  Numerals,  Largely  from  Works  on 

the  Abacus1 

12345G  789  0 

*    T  <f  L  A   8  j> 

gl  ar  rh  fi  1/  in  v<§  fe 

h  J  's-  !£  ^  <t  ?    ^\g    9    * 

I   ^   ji  /#    &    lb    V     &   S     <S> 


a  Friedlein  ed.,  p.  397.  b  Carlsruhe  codex  of  Gerlando. 

c  Munich  codex  of  Gerlando.         d  Carlsruhe  codex  of  Bernelinus. 

e  Munich  codex  of  Bernelinus.       f  Turchill,  c.  1200. 

e  Anon.  MS.,  thirteenth  century,  Alexandrian  Library,  Rome. 

h  Twelfth-century  Boethius,  Friedlein,  p.  396. 

>  Vatican  codex,  tenth  century,  Boethius. 

1  a,  h,  >,  are  from  the  Friedlein  ed.;  the  original  in  the  manuscript 
from  which  a  is  taken  contains  a  zero  symbol,  as  do  all  of  the  six 
plates  given  by  Friedlein.  b-e  from  the  Boncompagni  BidleUno,  Vol. 
X,  p.  59(5 ;  f  ibid.,  Vol.  XV,  p.  130  ;  e  Memorie  della  classe  di  sci.,  Eeale 
Ace.  dei  Lincei,  An.  CCLXXIV  (1876-1877),  April,  1877.  A  twelfth- 
century  arithmetician,  possibly  John  of  Luna  (Ilispalensis,  of  Seville, 
c.  1150),  speaks  of  the  great  diversity  of  these  forms  even  in  his  day, 
saying:  "Est  autem  in  aliquibus  figuram  istarum  apud  multos  diuer- 
sitas.  Quidam  cnim  septimam  banc  figuram  representant  .</.,  alii 
autem  sic  .^ty.,  uel  sic  A  .  Quidam  vero  quartam  sic  <>  ."  [Boncom- 
pagni, Trattati,  Vol.  II,  p.  28.] 


THE  BOETHIUS  QUESTION  89 

for  the  rest  of  his  discussion,  a  proceeding  so  foreign  to 
the  method  of  Boethius  as  to  be  inexplicable  on  the 
hypothesis  of  authenticity.  Why  should  such  a  scholarly 
writer  have  given  them  with  no  mention  of  their  origin 
or  use  ?  Either  he  would  have  mentioned  some  histor- 
ical interest  attaching  to  them,  or  he  would  have  used 
them  in  some  discussion ;  he  certainly  would  not  have 
left  the  passage  as  it  is. 

Sir  E.  Clive  Bayley  has  added  1  a  further  reason  for 
believing  them  spurious,  namely  that  the  4  is  not  of  the 
Nana  Ghat  type,  but  of  the  Kabul  form  which  the  Arabs 
did  not  receive  until  776  ; 2  so  that  it  is  not  likely,  even 
if  the  characters  were  known  in  Europe  in  the  time  of 
Boethius,  that  this  particular  form  was  recognized.  It 
is  worthy  of  mention,  also,  that  in  the  six  abacus  forms 
from  the  chief  manuscripts  as  given  by  Friedlein,3  each 
contains  some  form  of  zero,  which  symbol  probably  origi- 
nated in  India  about  this  time  or  later.  It  could  hardly 
have  reached  Europe  so  soon. 

As  to  the  fourth  question,  Did  Boethius  probably  know 
the  numerals  ?  It  seems  to  be  a  fair  conclusion,  accord- 
ing to  our  present  evidence,  that  (1)  Boethius  might 
very  easily  have  known  these  numerals  without  the  zero, 
but,  (2)  there  is  no  reliable  evidence  that  he  did  know 
them.  And  just  as  Boethius  might  have  come  in  contact 
with  them,  so  any  other  inquiring  mind  might  have  done 
so  either  in  his  time  or  at  any  time  before  they  definitely 
appeared  in  the  tenth  century.  These  centuries,  five  in 
number,  represented  the  darkest  of  the  Dark  Ages,  and 
even  if  these  numerals  were  occasionally  met  and  studied, 
no  trace  of  them  would  be  likely  to  show  itself  in  the 

1  Loc.  cit.,  p.  59.  2  Ibid.,  p.  101.  3  Loc.  cit.,  p.  396. 


90  THE  HINDU-ARABIC  NUMERALS 

literature  of  the  period,  unless  by  chance  it  should  get 
into  the  writings  of  some  man  like  Alcuin.    As  a  matter 
of  fact,  it  was  not  until  the  ninth  or  tenth  century  that 
there  is  any  tangible  evidence  of  their  presence  in  Chris- 
tendom.   They  were  probably  known  to  merchants  here 
and  there,  but  in  their  incomplete  state  they  were  not  of 
sufficient  importance  to  attract  any  considerable  attention. 
As  a  result  of  this  brief  survey  of  the  evidence  several 
conclusions  seem  reasonable :  (1)  commerce,  and  travel 
for  travel's  sake,  never  died  out  between  the  East  and  the 
West;  (2)  merchants  had  every  opportunity  of  knowing, 
and  would  have  been  unreasonably  stupid  if  they  had 
not  known,  the  elementary  number  systems  of  the  peo- 
ples with  whom  they  were  trading,  but  they  would  not 
have  put  tins  knowledge  in  permanent  written  form; 
(3)  wandering  scholars  would  have  known  many  and 
strange  things  about  the  peoples  they  met,  but  they  too 
were  not,  as  a  class,  writers ;  (4)  there  is  every  reason 
a  priori  for  believing  that  the  gobar  numerals  would 
have  been  known  to  merchants,  and  probably  to  some  of 
the  wandering  scholars,  long  before  the  Arabs  conquered 
northern  Africa ;   (5)  the  wonder  is  not  that  the  Hindu- 
Arabic  numerals  were  known  about  1000  A.D.,  and' that 
they  were  the  subject  of  an  elaborate  work  in  1202  by 
Fibonacci,  but  rather  that  more  extended  manuscript  evi- 
dence of  their  appearance  before  that  time  has  not  been 
found.    That  they  were  more  or  less  known  early  in  the 
Middle  Ages,  certainly  to  many  merchants  of  Christian 
Europe,  and  probably  to  several  scholars,  but  without 
the  zero,  is  hardly  to  be  doubted.    The  lack  of  docu- 
mentary evidence  is  not  at  all  strange,  in  view  of  all 
of  the  circumstances. 


CHAPTER  VI 

THE  DEVELOPMENT  OF  THE  NUMERALS 
AMONG  THE  ARABS 

If  the  numerals  had  their  origin  in  India,  as  seems 
most  probable,  when  did  the  Arabs  come  to  know  of 
them  ?  It  is  customary  to  say  that  it  was  due  to  the  in- 
fluence of  Mohammedanism  that  learning  spread  through 
Persia  and  Arabia ;  and  so  it  was,  in  part.  But  learning 
was  already  respected  in  these  countries  long  before  Mo- 
hammed appeared,  and  commerce  flourished  all  through 
this  region.  In  Persia,  for  example,  the  reign  of  Khosru 
Nuslrwan,1  the  great  contemporary  of  Justinian  the  law- 
maker, was  characterized  not  only  by  an  improvement  in 
social  and  economic  conditions,  but  by  the  cultivation  of 
letters.  Khosru  fostered  learning,  inviting  to  his  court 
scholars  from  Greece,  and  encouraging  the  introduction 
of  culture  from  the  West  as  well  as  from  the  East.  At 
this  time  Aristotle  and  Plato  were  translated,  and  por- 
tions of  the  H-ito-padesa,  or  Fables  of  Pilpay,  were  ren- 
dered from  the  Sanskrit  into  Persian.  All  this  means 
that  some  three  centuries  before  the  great  intellectual 
ascendancy  of  Bagdad  a  similar  fostering  of  learning  was 
taking  place  in  Persia,  and  under  pre-Mohammedan 
influences. 

1  Khosru  I,  who  began  to  reign  in  531  a.d.  See  W.  S.  W.  Vaux, 
Persia,  London,  1875,  p.  169;  Th.  Noldeke,  Aufsdtze  zur  persischen 
Geschichte,  Leipzig,  1887,  p.  113,  and  his  article  in  the  ninth  edition 
of  the  Encyclopuedia  Britannica. 

91 


92  THE  HINDU-ARABIC  NUMERALS 

■/-  The  first  definite  trace  that  we  have  of  the  introduc- 
tion of  the  Hindu  system  into  Arabia  dates  from  773  a.d.,1 
when  an  Indian  astronomer  visited  the  court  of  the  ca- 
liph, bringing  with  him  astronomical  tables  which  at  the 
caliph's  command  were  translated  into  Arabic  by  Al- 
Fazari.2  Al-Khowarazmi  and  Habash  (Ahmed  ibn  'Ab- 
dallah,  died  c.  870)  based  their  well-known  tables  upon 
the  work  of  Al-Fazari.  It  may  be  asserted  as  highly 
probable  that  the  numerals  came  at  the  same  time  as  the 
tables.  They  were  certainly  known  a  few  decades  later, 
and  before  825  a.d.,  about  which  time  the  original  of  the 
Algoritmi  de  numero  Indorum  was  written,  as  that  work 
makes  no  pretense  of  being  the  first  work  to  treat  of  the 
Hindu  numerals. 

The  three  writers  mentioned  cover  the  period  from  the 
end  of  the  eighth  to  the  end  of  the  ninth  century.  While 
the  historians  Al-Mas'udi  and  Al-Biruni  follow  quite 
closely  upon  the  men  mentioned,  it  is  well  to  note  again 
the  Arab  writers  on  Hindu  arithmetic,  contemporary  with 
Al-Khowaraznri,  who  were  mentioned  in  chapter  I,  viz. 
Al-Kindi,  Sened  ibn  'All,  and  Al-Sufi. 

For  over  five  hundred  years  Arabic  writers  and  others 
continued  to  apply  to  works  on  arithmetic  the  name 
"  Indian."  In  the  tenth  century  such  writers  are  'Abdal- 
lah  ibn  al-Hasan,  Abu  '1-Qasim3  (died  987  a.d.)  of  An- 
tioch,  and  Mohammed  ibn  'Abdallah,  Abu  Nasr 4  (c.  982), 
of  Kalwada  near  Bagdad.    Others  of  the  same  period  or 

1  Colebrooke,  Essays,  Vol.  II,  p.  504,  on  the  authority  of  Ibn  al- 
Adami,  astronomer,  in  a  work  published  by  his  continuator  Al-Qasim 
in  920  a.d. ;  Al-BIruni,  India,  Vol.  II,  p.  15. 

2  H.  Suter,  Die  Mathematiker  etc.,  pp.  4-5,  states  that  Al-Fazaii 
died  between  796  and  806. 

8  Suter,  loc.  cit.,  p.  63. 
4  Suter,  loc.  cit.,  p.  74. 


DEVELOPMENT  OF  THE  NUMERALS  93 

earlier  (since  they  are  mentioned  in  the  Fihrist,1  987a.d.), 
who  explicitly  use  the  word  "  Hindu  "  or  "  Indian,"  are 
Sinan  ibn  al-Fath2  of  Harran,  and  Ahmed  ibn  'Omar, 
al-Karabisi.3  In  the  eleventh  century  come  Al-Biruni4 
(973-1048)  and  'Ali  ibn  Ahmed,  Abu  '1-Hasan,  Al- 
Nasawi5  (c.  1030).  The  following  century  brings  simi- 
lar works  by  Ishaq  ibn  Yusuf  al-Sardafi6  and  Samu'Il 
ibn  Yahya  ibn  'Abbas  al-Magrebi  al-Andalusi7  (c.  1174), 
and  in  the  thirteenth  century  are  'Abdallatif  ibn  Yusuf 
ibn  Mohammed,  Muwaffaq  al-Din  Abu  Mohammed  al- 
Bagdadi8  (c.  1231),  and  Ibn  al-Banna.9 

The  Greek  monk  Maximus  Planudes,  writing  in  the 
first  half  of  the  fourteenth  century,  followed  the  Arabic 
usage  in  calling  his  work  Indian  Arithmetic. lQ  There  were 
numerous  other  Arabic  writers  upon  arithmetic,  as  that 
subject  occupied  one  of  the  high  places  among  the  sciences, 
but  most  of  them  did  not  feel  it  necessary  to  refer  to  the 
origin  of  the  symbols,  the  knowledge  of  which  might  well 
have  been  taken  for  granted. 

1  Suter,  Dots  Mathematiker-Verzeichniss  im  Fihrist.  The  references 
to  Suter,  unless  otherwise  stated,  are  to  his  later  work  Die  Mathemati- 
ker  und  Astronomen  der  Araber  etc. 

2  Suter,  Fihrist,  p.  37,  no  date.  f 

3  Suter,  Fihrist,  p.  38,  no  date.  / 

4  Possibly  late  tenth,  since  he  refers  to  one  arithmetical  work  which 
is  entitled  Book  of  the  Cyphers  in  his  Chronology,  English  ed.,  p.  132. 
Suter,  Die  Mathematiker  etc.,  pp.  98-100,  does  not  mention  this  work ; 
see  the  Nachtrdge  und  Berichtigungen,  pp.  170-172. 

5  Suter,  pp.  96-97. 

6  Suter,  p.  111. 

7  Suter,  p.  124.    As  the  name  shows,  he  came  from  the  West. 

8  Suter,  p.  138. 

9  Hankel,  Zur  Geschichte  der  Mathematik,  p.  256,  refers  to  him  as 
writing  on  the  Hindu  art  of  reckoning ;  Suter,  p.  162. 

10  tyy<po(popla.  kclt'  'ivdovs,  Greek  ed.,  C.  I.  Gerhardt,  Halle,  1865; 
and  German  translation,  Das  Iiechenbuch  des  Maximus  Planudes,  H. 
>Vaschke,  Halle,  1878. 


94  THE  HINDU-ARABIC  NUMERALS 

One  document,  cited  by  Woepcke,1  is  of  special  inter- 
est since  it  shows  at  an  early  period,  970  A.D.,  the  use 
of  the  ordinary  Arabic  forms  alongside  the  gobar.  The 
title  of  the  work  is  Interesting  and  Beautiful  Problems  on 
Numbers  copied  by  Ahmed  ibn  Mohammed  ibn  'Abdaljalil, 
Abu  Sa'Id,  al-Sijzi,2  (951-1024)  from  a  work  by  a  priest 
and  physician,  Nazif  ibn  Yumn,3  al-Qass  (died  c.  990). 
Suter  does  not  mention  this  work  of  Nazif. 

The  second  reason  for  not  ascribing  too  much  credit 
to  the  purely  Arab  influence  is  that  the  Arab  by  himself 
never  showed  any  intellectual  strength.  What  took  place 
after  Mohammed  had  lighted  the  fire  in  the  hearts  of  his 
people  was  just  what  always  takes  place  when  different 
types  of  strong  races  blend,  —  a  great  renaissance  in 
divers  lines,  It  was  seen  in  the  blending  of  such  types  at 
Miletus  in  the  time  of  Thales,  at  Rome  in  the  days  of 
the  early  invaders,  at  Alexandria  when  the  Greek  set 
firm  foot  on  Egyptian  soil,  and  we  see  it  now  when  all 
the  nations  mingle  their  vitality  in  the  New  World.  So 
when  the  Arab  culture  joined  with  the  Persian,  a  new 
civilization  rose  and  flourished.4  The  Arab  influence 
came  not  from  its  purity,  but  from  its  intermingling  with 
an  influence  more  cultured  if  less  virile. 

As  a  result  of  this  interactivity  among  peoples  of  diverse 

interests  and  powers,  Mohammedanism  was  to  the  world 

/  from  the  eighth  to  the  thirteenth  century  what  Rome  and 

Athens  and  the  Italo-Hellenic  influence  generally  had 

1  "  Sur  une  <lonn£e  historique  relative  a  Temploi  des  chiffres  in- 
diens  par  les  Arabes,'"  Tortolini's  Annali  di  scienze  mat.  efis.,  1855. 

2  Suter,  p.  80. 
8  Suter,  p.  08. 

4  Sprenger  also  calls  attention  to  this  fact,  in  tbe  Ztitsdirift  d. 
deutschen  muryaddud.  Gvscllxcliaft,  Vol.  XLV.  p.  307. 


DEVELOPMENT  OF  THE  NUMERALS  95 

been  to  the  ancient  civilization.  "  If  they  did  not  possess 
the  spirit  of  invention  which  distinguished  the  Greeks 
and  the  Hindus,  if  they  did  not  show  the  perseverance 
in  their  observations  that  characterized  the  Chinese 
astronomers,  they  at  least  possessed  the  virility  of  a  new 
and  victorious  people,  with  a  desire  to  understand  what 
others  had  accomplished,  and  a  taste  which  led  them 
with  equal  ardor  to  the  study  of  algebra  and  of  poetry, 
of  philosophy  and  of  language."  1 

It  was  in  622  a.d.  that  Mohammed  fled  from  Mecca, 
and  within  a  century  from  that  time  the  crescent  had 
replaced  the  cross  in  Christian  Asia,  in  Northern  Africa, 
and  in  a  goodly  portion  of  Spain.  The  Arab  empire  was 
an  ellipse  of  learning  with  its  foci  at  Bagdad  and  Cor- 
dova, and  its  rulers  not  infrequently  took  pride  in  de- 
manding intellectual  rather  than  commercial  treasure  as 
the  result  of  conquest.2 

It  was  under  these  influences,  either  pre-Mohammedan 
or  later,  that  the  Hindu  numerals  found  their  way  to  the^ 
North.  If  they  were  known  before  Mohammed's  time, 
the  proof  of  this  fact  is  now  lost.  This  much,  however, 
is  known,  that  in  the  eighth  century  they  were  taken  to 
Bagdad.  It  was  early  in  that  century  that  the  Moham- 
medans obtained  their  first  foothold  in  northern  India, 
thus  foreshadowing  an  epoch  of  supremacy  that  endured 
with  varied  fortunes  until  after  the  golden  age  of  Akbar 
the  Great  (1542-1605)  and  Shah  Jehan.  They  also  con- 
quered Khorassan  and  Afghanistan,  so  that  the  learning 
and  the  commercial  customs  of  India  at  once  found  easy 

1  Libri,  Histoire  des  mathematiques.  Vol.  I.  p.  147. 

2  "Dictant  la  paix  a  Tempereur  de  Constantinople.  PArabe  victo- 
rieux  demandait  des  rnannscrits  et  des  savans."  [Libri.  loc.  cit., 
p.  108.] 


96  THE  HINDU-ARABIC  NUMERALS 

access  to  the  newly-established  schools  and  the  bazaars  of 
Mesopotamia  and  western  Asia.  The  particular  paths  of 
conquest  and  of  commerce  were  either  by  way  of  the 
Khyber  Pass  and  through  Kabul,  Herat  and  Khorassan, 
or  by  sea  through  the  strait  of  Ormuz  to  Basra  (Busra) 
at  the  head  of  the  Persian  Gulf,  and  thence  to  Bagdad. 
As  a  matter  of  fact,  one  form  of  Arabic  numerals,  the  one 
now  in  use  by  the  Arabs,  is  attributed  to  the  influence  of 
Kabul,  while  the  other,  which  eventually  became  our  nu- 
merals, may  very  likely  have  reached  Arabia  by  the  other 
route.  It  is  in  Bagdad,1  Dar  al-Salam  —  "  the  Abode  of 
Peace,"  that  our  special  interest  in  the  introduction  of  the 
numerals  centers.  Built  upon  the  ruins  of  an  ancient 
town  by  Al-Mansur2  in  the  second  half  of  the  eighth 
century,  it  lies  in  one  of  those  regions  where  the  converg- 
ing routes  of  trade  give  rise  to  large  cities.3  Quite  as 
well  of  Bagdad  as  of  Athens  might  Cardinal  Newman 
have  said : 4 

"What  it  lost  in  conveniences  of  approach,  it  gained 
in  its  neighborhood  to  the  traditions  of  the  mysterious 
East,  and  in  the  loveliness  of  the  region  in  which  it  lay. 
Hither,  then,  as  to  a  sort  of  ideal  land,  where  all  arche- 
types of  the  great  and  the  fair  were  found  in  substantial 
being,  and  all  departments  of  truth  explored,  and  all 
diversities  of  intellectual  power  exhibited,  where  taste 
and  philosophy  were  majestically  enthroned  as  in  a  royal 
court,  where  there  was  no  sovereignty  but  that  of  mind, 
and  no  nobility  but  that  of  genius,  where  professors  were 

1  Persian  bagadata,  "God-given." 

2  One  of  the  Abbassides,  the  (at  least  pretended)  descendants  of 
'A  1- Abbas,  uncle  and  adviser  of  Mohammed. 

:s  E.  Reclus,  Asia,  American  ed.,N.Y.,  1891,  Vol.  IV,  p.  227. 
4  Historical  Sketches,  Vol.  HI,  chap.  iii. 


DEVELOPMENT  OF  THE  NUMERALS  97 

rulers,  and  princes  did  homage,  thither  nocked  continually 
from  the  very  corners  of  the  orbis  terrarum  the  many- 
tongued  generation,  just  rising,  or  just  risen  into  man- 
hood, in  order  to  gain  wisdom."  For  here  it  was  that 
Al-Mansur  and  Al-Mamun  and  Harun  al-Rashld  (Aaron 
the  Just)  made  for  a  time  the  world's  center  of  intellec- 
tual "activity  in  general  and  in  the  domain  of  mathematics 
in  particular.1  It  was  just  after  the  Sindhind  was  brought 
to  Bagdad  that  Mohammed  ibn  Musa  al-Khowarazml, 
whose  name  has  already  been  mentioned,2  was  called  to 
that  city.  He  was  the  most  celebrated  mathematician  of 
his  time,  either  in  the  East  or  West,  writing  treatises  on 
arithmetic,  the  sundial,  the  astrolabe,  chronology,  geom- 
etry, and  algebra,  and  giving  through  the  Latin  translit- 
eration of  his  name,  algoritmi,  the  name  of  algorism  to  the 
early  arithmetics  using  the  new  Hindu  numerals.3  Appre- 
ciating at  once  the  value  of  the  position  system  so  recently 
brought  from  India,  he  wrote  an  arithmetic  based  upon 
these  numerals,  and  this  was  translated  into  Latin  in  the 
time  of  Adelhard  of  Bath  (c.  1130),  although  possibly  by 
his  contemporary  countryman  Robert  Cestrensis.4  This 
translation  was  found  in  Cambridge  and  was  published 
by  Boncompagni  in  1857.5 

Contemporary  with  Al-Khowarazml,  and  working  also 
under  Al-Mamun,  was  a  Jewish  astronomer,  Abu  '1-Teiyib, 

1  On  its  prominence  at  that  period  see  Villicus,  p.  70. 

2  See  pp.  4-5. 

3  Smith,  D.  E.,  in  the  Cantor  Festschrift,  1909,  note  pp.  10-11.  See 
also  F.  Woepcke,  Propagation. 

4  Enestrom,  in  Bibliotheca  Mathematica,  Vol.  I  (3),  p.  499 ;  Cantor, 
Geschichte,  Vol.  1(3),  p.  671. 

5  Cited  in  Chapter  I.  It  begins:  "Dixit  algoritmi :  laudes  deo  rec- 
tori  nostro  atque  defensori  dicamus  dignas."  It  is  devoted  entirely 
to  the  fundamental  operations  and  contains  no  applications, 


sj 


98  THE  HINDU-ARABIC  NUMERALS 

Sened  ibn  'All,  who  is  said  to  have  adopted  the  Moham- 
medan religion  at  the  caliph's  request.  He  also  wrote  a 
work  on  Hindu  arithmetic,1  so  that  the  subject  must  have 
been  attracting  considerable  attention  at  that  time.  In- 
deed, the  struggle  to  have  the  Hindu  numerals  replace 
the  Arabic  did  not  cease  for  a  long  time  thereafter.  'All 
ibn  Ahmed  al-Nasawi,  in  his  arithmetic  of  c.  1025,* tells 
us  that  the  symbolism  of  number  was  still  unsettled  in 
his  day,  although  most  people  preferred  the  strictly 
Arabic  forms.2 

We  thus  have  the  numerals  in  Arabia,  in  two  forms : 
one  the  form  now  used  there,  and  the  other  the  one  used 
by  Al-Khowarazmi.  The  question  then  remains,  how  did 
this  second  form  find  its  way  into  Europe  ?  and  this  ques- 
tion will  be  considered  in  the  next  chapter. 

1  M.  Steinschneider,  "Die  Mathematik  bei  den  Jude^1'  Bibliotheca 
Mathematica,  Vol.  VIII  (2),  p.  99.  See  also  the  reference  to  this  writer 
in  Chapter  I. 

2  Part  of  this  work  has  been  translated  from  a  Leyden  MS.  by  F. 
Woepcke,  Propagation,  and  more  recently  by  H.  Suter,  Bibliotheca 
Mathtmatica,Yo\.  VII (3),  pp.  113-119. 


CHAPTER  VII 

THE  DEFINITE  INTRODUCTION  OF  THE  NUMERALS 

INTO  EUROPE 

It  being  doubtful  whether  Boethius  ever  knew  the 
Hindu  numeral  forms,  certainly  without  the  zero  in  any 
case,  it  becomes  necessary  now  to  consider  the  question 
of  their  definite  introduction  into  Europe.  From  what 
has  been  said  of  the  trade  relations  between  the  East  and 
the  West,  and  of  the  probability  that  it  was  the  trader 
rather  than  the  scholar  who  carried  these  numerals  from 
their  original  habitat  to  various  commercial  centers,  it  is 
evident  that  we  shall  never  know  when  they  first  made 
then  inconspicuous  entrance  into  Europe.  Curious  cus- 
toms from  the  East  and  from  the  tropics,  —  concerning 
games,  social  peculiarities,  oddities  of  dress,  and  the  like, 
- —  are  continually  being  related  by  sailors  and  traders  in 
their  resorts  in  New  York,  London,  Hamburg,  and  Rot- 
terdam to-day,  customs  that  no  scholar  has  yet  described 
in  print  and  that  may  not  become  known  for  many  years, 
if  ever.  And  if  this  be  so  now,  how  much  more  would  it 
have  been  true  a  thousand  years  before  the  invention  of 
printing,  when  learning  was  at  its  lowest  ebb.  It  was  at 
this  period  of  low  esteem  of  culture  that  the  Hindu  numer- 
als undoubtedly  made  their  first  appearance  in  Europe. 

There  were  many  opportunities  for  such  knowledge  to 
reach  Spain  and  Italy.  In  the  first  place  the  Moors  went 
into  Spain  as  helpers  of  a  claimant  of  the  throne,  and 

99 


100  THE  HINDU-ARABIC  NUMERALS 

remained  as  conquerors.  The  power  of  the  Goths,  who 
had  held  Spain  for  three  centuries,  was  shattered  at  the 
battle  of  Jerez  de  la  Frontera  in  711,  and  almost  imme- 
diately the  Moors  became  masters  of  Spain  and  so  re- 

Vmained  for  five  hundred  years,  and  masters  of  Granada 
for  a  much  longer  period.  Until  850  the  Christians  were 
absolutely  free  as  to  religion  and  as  to  holding  political 
office,  so  that  priests  and  monks  were  not  infrequently 
skilled  both  in  Latin  and  Arabic,  acting  as  official  trans- 
lators, and  naturally  reporting  directly  or  indirectly  to 
Rome.  There  was  indeed  at  this  time  a  complaint  that 
Christian  youths  cultivated  too  assiduously  a  love  for 
the  literature  of  the  Saracen,  and  married  too  frequently 
the  daughters  of  the  infidel.1  It  is  true  that  this  happy 
state  of  affairs  was  not  permanent,  but  while  it  lasted 
the  learning  and  the  customs  of  the  East  must  have  be- 
come more  or  less  the  property  of  Christian  Spain.  At 
this  time  the  gobar  numerals  were  probably  in  that  coun- 
try, and  these  may  well  have  made  their  way  into  Europe 

ffrom  the  schools  of  Cordova,  Granada,  and  Toledo. 

Furthermore,  there  was  abundant  opportunity  for  the 
numerals  of  the  East  to  reach  Europe  through  the  jour- 
neys of  travelers  and  ambassadors.  It  was  from  the  rec- 
ords of  Suleiman  the  Merchant,  a  well-known  Arab  trader 
of  the  ninth  century,  that  part  of  the  story  of  Sindbad 
the  Sailor  was  taken.2  Such  a  merchant  would  have  been 
particularly  likely  to  know  the  numerals  of  the  people 
whom  he  met,  and  he  is  a  type  of  man  that  may  well  have 
taken  such  symbols  to  European  markets.  A  little  later, 

1  A.  Neander,  General  History  of  the  Christian  Religion  and  Church, 
5th  American  ed.,  Boston,  1855,  Vol.  Ill,  p.  335. 

2  Beazley,  loc.  cit.,  Vol.  I,  p.  49. 


DEFINITE  INTRODUCTION  INTO  EUfcOi'Iy     Itffc 

Abu  '1-Hasan  rAH  al-Mas'udi  (d.  956)  of  Bagdad  traveled 
to  the  China  Sea  on  the  east,  at  least  as  far  south  as 
Zanzibar,  and  to  the  Atlantic  on  the  west,1  and  he  speaks 
of  the  nine  figures  with  which  the  Hindus  reckoned.2 

There  was  also  a  Bagdad  merchant,  one  Abu  '1-Qasim 
'Obeidallah  ibn  Ahmed,  better  known  by  his  Persian 
name  Ibn  Khordadbeh,3  who  wrote  about  850  A.D.  a 
work  entitled  Book  of  Mo  ads  and  Provinces  4  in  which  the 
following  graphic  account  appears  : 5  "  The  Jewish  mer- 
^_  chants  speak  Persian,  Roman  (Greek  and  Latin),  Arabic, 
French,  Spanish,  and  Slavic.  They  travel  from  the  West 
to  the  East,  and  from  the  East  to  the  West,  sometimes 
by  land,  sometimes  by  sea.  They  take  ship  from  France 
on  the  Western  Sea,  and  they  voyage  to  Farama  (near 
the  ruins  of  the  ancient  Pelusium)  ;  there  they  transfer 
their  goods  to  caravans  and  go  by  land  to  Colzom  (on  the 
Red  Sea).  They  there  reembark  on  the  Oriental  (Red) 
Sea  and  go  to  Hejaz  and  to  Jiddah,  and  thence  to  the 
Sind,  India,  and  China.  Returning,  they  bring  back  the 
products  of  the  oriental  lands.  .  .  .  These  journeys  are 
also  made  by  land.  The  merchants,  leaving  France  and 
Spain,  cross  to  Tangier  and  thence  pass  through  the 
African  provinces  and  Egypt.  They  then  go  to  Ram- 
leh,  visit  Damascus,  Kufa,  Bagdad,  and  Basra,  penetrate 
into  Ahwaz,  Fars,  Kerman,  Sind,  and  thus  reach  India 
and  China."  Such  travelers,  about  900  a.d.,  must  neces- 
sarily have  spread  abroad  a  knowledge  of  all  number 

1  Beazley,  loc.  cit.,  Vol.  I,  pp.  50,  460. 

2  See  pp.  7-8. 

3  The  name  also  appears  as  Mohammed  Abu*l-Qasim,  and  Ibn  Hau- 
qal.    Beazley,  loc.  cit.,  Vol.  I,  p.  45. 

4  Kitab  al-masalik  wcCl-mamalik. 

5  Reinaud,  Mim.  sur  VInde;  in  Gerhardt,  Etudes,  p.  18. 


102  THE  JilNDU-ARABIC  NUMERALS 

systems  used  in  recording  prices  or  in  the  computations 
of  the  market.  There  is  an  interesting  witness  to  this 
movement,  a  cruciform  brooch  now  in  the  British  Mu- 
seum. It  is  English,  certainly  as  early  as  the  eleventh 
century,  but  it  is  inlaid  with  a  piece  of  paste  on  which 
is  the  Mohammedan  inscription,  in  Kufic  characters, 
"  There  is  no  God  but  God."  How  did  such  an  inscrip- 
tion find  its  way,  perhaps  in  the  time  of  Alcuin  of  York, 
to  England  ?  And  if  these  Kufic  characters  reached 
there,  then  why  not  the  numeral  forms  as  well  ? 

Even  in  literature  of  the  better  class  there  appears 
now  and  then  some  stray  proof  of  the  important  fact 
that  the  great  trade  routes  to  the  far  East  were  never 
closed  for  long,  and  that  the  customs  and  marks  of  trade 
endured  from  generation  to  generation.  The  Grtrtutan  of 
the  Persian  poet  Sa'di l  contains  such  a  passage : 

"  I  met  a  merchant  who  owned  one  hundred  and  forty 
camels,  and  fifty  slaves  and  porters.  .  .  .  He  answered  to 
me :  '  I  want  to  carry  sulphur  of  Persia  to  China,  which 
in  that  country,  as  I  hear,  bears  a  high  price  ;  and  thence 
to  take  Chinese  ware  to  Roum  ;  and  from  Roum  to  load 
up  with  brocades  for  Hind  ;  and  so  to  trade  Indian  steel 
(jydlafr)  to  Halib.  From  Halib  I  will  convey  its  glass  to 
Yeman,  and  carry  the  painted  cloths  of  Yeman  back  to 
Persia.'  "  2  On  the  other  hand,  these  men  were  not  of  the 
learned  class,  nor  would  they  preserve  in  treatises  any 
knowledge  that  they  might  have,  although  this  knowl- 
edge would  occasionally  reach  the  ears  of  the  learned  as 
bits  of  curious  information. 

1  Born  at  Sliiraz  in  1193.  He  himself  had  traveled  from  India 
to  Europe. 

2  Gulistan  (Rose  Garden),  Gateway  the  third,  XXII.  Sir  Edwin 
Arnold's  translation,  N.Y.,  1899,  p.  177. 


DEFINITE  INTRODUCTION  INTO  EUROPE      103 

There  were  also  ambassadors  passing  back  and  forth 
from  time  to  time,  between  the  East  and  the  West,  and 
in  particular  during  the  period  when  these  numerals 
probably  began  to  enter  Europe.  Thus  Charlemagne 
(c.  800)  sent  emissaries  to  Bagdad  just  at  the  time  of 
the  opening  of  the  mathematical  activity  there.1  And 
with  such  ambassadors  must  have  gone  the  adventurous 
scholar,  inspired,  as  Alcuin  says  of  Archbishop  Albert 
of  York  (70 6-780), 2  to  seek  the  learning  of  other  lands. 
Furthermore,  the  Nestorian  communities,  established  in 
Eastern  Asia  and  in  India  at  this  time,  were  favored  both 
by  the  Persians  and  by  their  Mohammedan  conquerors. 
The  Nestorian  Patriarch  of  Syria,  Timotheus  (778-820), 
sent  missionaries  both  to  India  and  to  China,  and  a  bishop 
was  appointed  for  the  latter  field.  Ibn  Wahab,  who  trav- 
eled to  China  in  the  ninth  century,  found  images  of  Christ 
and  the  apostles  in  the  Emperors  court.3  Such  a  learned 
body  of  men,  knowing  intimately  the  countries  in  which 
they  labored,  could  hardly  have  failed  to  make  strange 
customs  known  as  they  returned  to  their  home  stations. 
Then,  too,  in  Alfred's  time  (849-901)  emissaries  went 

1  Cunningham,  loc.  cit.,  p.  81. 

2  Putnam,  Books,  Vol.  I,  p.  227  : 

"  Non  semel  externas  peregrino  tramite  terras 
Jam  peragravit  ovans,  sophiae  deductus  amore, 
Si  quid  forte  novi  librorum  seii  studiormn 
Quod  securn  ferret,  terris  reperiret  in  illis. 
Hie  quoque  Romuleum  venit  devotus  ad  urbem." 

("  More  than  once  he  has  traveled  joyfully  through  remote  regions  and 
by  strange  roads,  led  on  by  his  zeal  for  knowledge  and  seeking  to  discover 
in  foreign  lands  novelties  in  books  or  in  studies  which  he  could  take  back 
with  him.    And  this  zealous  student  journeyed  to  the  city  of  Romulus.") 

3  A.  Neander,  General  History  of  the  Christian  Religion  and  Church, 
5th  American  ed.,  Boston,  1855,  Vol.  Ill,  p.  89,  note  4 ;  Libri,  Histoire, 
Vol.  I,  p.  143. 


104  THE  HINDU-ARABIC  NUMERALS 

from  England  as  far  as  India,1  and  generally  in  the 
Middle  Ages  groceries  came  to  Europe  from  Asia  as  now 
they  come  from  the  colonies  and  from  America.  Syria, 
Asia  Minor,  and  Cyprus  furnished  sugar  and  wool,  and 
India  yielded  her  perfumes  and  spices,  while  rich  tapes- 
tries for  the  courts  and  the  wealthy  burghers  came  from 
Persia  and  from  China.2  Even  in  the  time  of  Justinian 
(c.  550)  there  seems  to  have  been  a  silk  trade  with  China, 
which  country  in  turn  carried  on  commerce  with  Ceylon,3 
and  reached  out  to  Turkestan  where  other  merchants 
transmitted  the  Eastern  products  westward.  In  the  sev- 
enth century  there  was  a  well-defined  commerce  between 
Persia  and  India,  as  well  as  between  Persia  and  Con- 
stantinople.4 The  Byzantine  commerciarii  were  stationed 
at  the  outposts  not  merely  as  customs  officers  but  as 
government  purchasing  agents.5 

Occasionally  there  went  along  these  routes  of  trade 
men  of  real  learning,  and  such  would  surely  have  carried 
the  knowledge  of  many  customs  back  and  forth.  Thus 
at  a  period  when  the  numerals  are  known  to  have  been 
partly  understood  hi  Italy,  at  the  opening  of  the  eleventh 
century,  one  Constantine,  an  African,  traveled  from  Italy 
through  a  great  part  of  Africa  and  Asia,  even  on  to 
India,  for  the  purpose  of  learning  the  sciences  of  the 
Orient.  He  spent  thirty-nine  years  in  travel,  having 
been  hospitably  received  in  Babylon,  and  upon  his  return 
he  was  welcomed  with  great  honor  at  Salerno.6 

A  very  interesting  illustration  of  this  intercourse  also 
appears  in  the  tenth  century,  when  the  son  of  Otto  I 

1  Cunningham,  loc.  cit.,  p.  81.      4  Ibid.,  p.  21. 

2  Heyd,  Inc.  cit.,  Vol.  I,  p.  4.        5  Ibid.,  p.  23. 

3  Ibid.,  p.  5.  6  Libri,  Ilistoire,  Vol.  I,  p.  167. 


DEFINITE  INTRODUCTION  INTO  EUROPE     105 

(936-973)  married  a  princess  from  Constantinople.  This 
monarch  was  in  touch  with  the  Moors  of  Spain  and 
invited  to  his  court  numerous  scholars  from  abroad,1 
and  his  intercourse  with  the  East  as  well  as  the  West 
must  have  brought  together  much  of  the  learning  of 
each. 

Another  powerful  means  for  the  circulation  of  mysti- 
cism and  philosophy,  and  more  or  less  of  culture,  took  its 
start  just  before  the  conversion  of  Constantine  (c.  312), 
in  the  form  of  Christian  pilgrim  travel.  This  was  a 
feature  peculiar  to  the  zealots  of  early  Christianity, 
found  in  only  a  slight  degree  among  their  Jewish  prede- 
cessors in  the  annual  pilgrimage  to  Jerusalem,  and 
almost  wholly  wanting  in  other  pre-Christian  peoples. 
Chief  among  these  early  pilgrims  were  the  two  Placen- 
tians,  John  and  Antonine  the  Elder  (c.  303),  who,  in 
their  wanderings  to  Jerusalem,  seem  to  have  started  a 
movement  which  culminated  centuries  later  in  the  cru- 
sades.2 In  333  a  Bordeaux  pilgrim  compiled  the  first 
Christian  guide-book,  the  Itinerary  from  Bordeaux  to 
Jerusalem?  and  from  this  time  on  the  holy  pilgrimage 
never  entirely  ceased. 

Still  another  certain  route  for  the  entrance  of  the  nu- 
merals into  Christian  Europe  was  through  the  pillaging 
and  trading  carried  on  by  the  Arabs  on  the  northern 
shores  of  the  Mediterranean.  As  early  as  652  a.d.,  in 
the  thirtieth  year  of  the  Hejira,  the  Mohammedans  de- 
scended upon  the  shores  of  Sicily  and  took  much  spoil. 
Hardly  had  the  wretched  Constans  given  place  to  the 

1  Picavet,  Gerbert,  un  pape  philosophe,  d'apres  Vhistoire  et  d'apres 
la  legende,  Paris,  1897,  p.  19. 

2  Beazley,  loc.  cit.,  Vol.  I,  chap,  i,  and  p.  54seq.  3  Ibid.,  p.  57. 


100  THE  HINDU-ARABIC  NUMERALS 

young  Constantine  IV  when  they  again  attacked  the 
island  and  plundered  ancient  Syracuse.  Again  in  827, 
under  A  sad,  they  ravaged  the  coasts.  Although  at  this 
time  they  failed  to  conquer  Syracuse,  they  soon  held  a 
good  part  of  the  island,  and  a  little  later  they  success- 
fully besieged  the  city.  Before  Syracuse  fell,  however, 
they  had  plundered  the  shores  of  Italy,  even  to  the  walls 
of  Rome  itself  ;  and  had  not  Leo  IV,  in  849,  repaired  the 
neglected  fortifications,  the  effects  of  the  Moslem  raid  of 
that  year  might  have  been  very  far-reaching.  Ibn  Khor- 
dadbeh,  who  left  Bagdad  in  the  latter  part  of  the  ninth 
century,  gives  a  picture  of  the  great  commercial  activity 
at  that  time  in  the  Saracen  city  of  Palermo.  In  this  same 
century  they  had  established  themselves  in  Piedmont, 
and  in  906  they  pillaged  Turin.1  On  the  Sorrento  pen- 
insula the  traveler  who  climbs  the  hill  to  the  beautiful 
Ravello  sees  still  several  traces  of  the  Arab  architecture, 
reminding  him  of  the  fact  that  about  900  a.d.  Amalfi  was 
a  commercial  center  of  the  Moors.2  Not  only  at  this  time, 
but  even  a  century  earlier,  the  artists  of  northern  India 
sold  their  wares  at  such  centers,  and  in  the  courts  both  of 
Harun  al-Rashid  and  of  Charlemagne.3  Thus  the  Arabs 
dominated  the  Mediterranean  Sea  long  before  Venice 

"  held  the  gorgeous  East  in  fee 
And  was  the  safeguard  of  the  West," 

and  long  before  Genoa  had  become  her  powerful  rival.4 

1  Libri,  Histoire,  Vol.1,  p.  110,  n.,  citing  authorities,  and  p.  152. 

-  Possibly  the  old  tradition,  "Prima  dedit  nautis  usuni  magnetis 
Amalphis,"  is  true  so  far  as  it  means  the  modern  form  of  compass 
card.    See  Beazley,  loc.  cit.,  Vol.11,  p.  398. 

■  K.  C.  Dutt,  loc.  cit.,  Vol.  II,  p.  312. 

4  E.  -J.  Payne,  in  The  Cambridge  Modern  History,  London,  11)02, 
Vol.  1.  chap.  i. 


DEFINITE  INTRODUCTION  INTO  EUROPE      107 

Only  a  little  later  than  this  the  brothers  Nicolo  and 
Maffeo  Polo  entered  upon  their  famous  wanderings.1 
Leaving  Constantinople  in  12(30,  they  went  by  the  Sea 
of  Azov  to  Bokhara,  and  thence  to  the  court  of  Kublai 
Khan,  penetrating  China,  and  returning  by  way  of  Acre 
in  1269  with  a  commission  which  required  them  to  go 
back  to  China  two  years  later.  This  time  they  took 
with  them  Nicolo's  son  Marco,  the  historian  of  the  jour- 
ney, and  went  across  the  plateau  of  Pamir ;  they  spent 
about  twenty  years  in  China,  and  came  back  by  sea  from 
China  to  Persia. 

The  ventures  of  the  Poli  were  not  long  unique,  how- 
ever: the  thirteenth  century  had  not  closed  before  Roman 
missionaries  and  the  merchant  Petrus  de  Lucolongo  had 
penetrated  China.  Before  1350  the  company  of  mission- 
aries was  large,  converts  were  numerous,  churches  and 
Franciscan  convents  had  been  organized  in  the  East, 
travelers  were  appealing  for  the  truth  of  their  accounts 
to  the  "many"  persons  in  Venice  who  had  been  in  China, 
Tsuan-chau-fu  had  a  European  merchant  community, 
and  Italian  trade  and  travel  to  China  was  a  thing  that 
occupied  two  chapters  of  a  commercial  handbook.2 

1  Geo.  Phillips,  "The  Identity  of  Marco  Polo's  Zaitun  with  Chang- 
chau,  in  T'oung  pao,"  Archives  pour  servir  a  V etude  de  Vhistoire  de 
VAsie  orientate,  Leyden,  1890,  Vol.  I,  p.  218.  W.  Heyd,  Geschichte  des 
Levanthandels  im  Mittelalter,  Vol.  II,  p.  210. 

The  Palazzo  dei  Poli,  where  Marco  was  born  and  died,  still  stands 
in  the  Corte  del  Milione,  in  Venice.  The  best  description  of  the  Polo 
travels,  and  of  other  travels  of  the  later  Middle  Ages,  is  found  in 
C.  R.  Beazley's  Dawn  of  Modern  Geography,  Vol.  Ill,  chap,  ii,  and 
Part  II. 

2  Heyd,  loc.  cit.,  Vol.  II,  p.  220  ;  H.  Yule,  in  Encyclopaedia  Britan- 
nica,  9th  (10th)  or  11th  ed.,  article  "China."  The  handbook  cited  is 
Pegolotti's  Libro  di  divisamenti  di  paesi,  chapters  i-ii,  where  it  is  im- 
plied that  $60,000  would  be  a  likely  amount  for  a  merchant  going  to 
China  to  invest  in  his  trip. 


108  THE  HINDU-ARABIC  NUMERALS 

It  is  therefore  reasonable  to  conclude  that  in  the  Mid- 
dle Ages,  as  in  the  time  of  Boethius,  it  was  a  simple 
matter  for  any  inquiring  scholar  to  become  acquainted 
with  such  numerals  of  the  Orient  as  merchants  may  have 
used  for  warehouse  or  price  marks.  And  the  fact  that 
Gerbert  seems  to  have  known  only  the  forms  of  the  sim- 
plest of  these,  not  comprehending  their  full  significance, 
seems  to  prove  that  he  picked  them  up  in  just  this  way. 

Even  if  Gerbert  did  not  bring  his  knowledge  of  the 
Oriental  numerals  from  Spain,  he  may  easily  have  ob- 
tained them  from  the  marks  on  merchant's  goods,  had  he 
been  so  inclined.  Such  knowledge  was  probably  ob- 
tainable hi  various  parts  of  Italy,  though  as  parts  of  mere 
mercantile  knowledge  the  forms  might  soon  have  been 
lost,  it  needing  the  pen  of  the  scholar  to  preserve  them. 
Trade  at  this  time  was  not  stagnant.  During  the  eleventh 
and  twelfth  centuries  the  Slavs,  for  example,  had  very 
great  commercial  interests,  their  trade  reaching  to  Kiev 
and  Novgorod,  and  thence  to  the  East.  Constantinople 
was  a  great  clearing-house  of  commerce  with  the  Orient,1 
and  the  Byzantine  merchants  must  have  been  entirely 
familiar  with  the  various  numerals  of  the  Eastern  peoples. 
In  the  eleventh  century  the  Italian  town  of  Amain  estab- 
lished a  factory'2  in  Constantinople,  and  had  trade  re- 
lations with  Antioch  and  Egypt.  Venice,  as  early  as  the 
ninth  century,  had  a  valuable  trade  with  Syria  and  Cairo.3 
Fifty  'years  after  Gerbert  died,  in  the  time  of  Cnut,  the 
Dane  and  the  Norwegian  pushed  their  commerce  far  be- 
yond the  northern  seas,  both  by  caravans  through  Russia 
to  the  Orient,  and  by  their  venturesome  barks  which 

1  Cunningham,  loc.  cit.,  p.  104.  -  I.e.  ;i  commission  house. 

3  Cunningham,  loc.  cit.,  p.  180. 


DEFINITE  INTRODUCTION  INTO  EUROPE      109 

sailed  through  the  Strait  of  Gibraltar  into  the  Medi- 
terranean.1 Only  a  little  later,  probably  before  1200  a.d., 
a  clerk  in  the  service  of  Thomas  a  Becket,  present  at  the 
latter  s  death,  wrote  a  life  of  the  martyr,  to  which  (fortu- 
nately for  our  purposes)  he  prefixed  a  brief  eulogy  of 
the  city  of  London.2  This  clerk,  William  Fitz  Stephen 
by  name,  thus  speaks  of  the  British  capital : 

Aurum  mittit  Arabs  :  species  et  thura  Sabseus  : 
Anna  Sythes  :  oleum  palmarum  divite  sylva 
Pingue  solum  Babylon  :  Nilus  lapides  pretiosos  : 
Norwegi,  Russi,  varium  grisum,  sabdinas  : 
Seres,  purpureas  vestes  :  Galli,  sua  vina. 

Although,  as  a  matter  of  fact,  the  Arabs  had  no  gold  to 
send,  and  the  Scythians  no  arms,  and  Egypt  no  precious 
stones  save  only  the  turquoise,  the  Chinese  (Seres)  may 
have  sent  their  purple  vestments,  and  the  north  her  sables 
and  other  furs,  and  France  her  wines.  At  any  rate  the 
verses  show  very  clearly  an  extensive  foreign  trade. 

Then  there  were  the  Crusades,  which  in  these  times 
brought  the  East  in  touch  with  the  West.  The  spirit  of 
the  Orient  showed  itself  in  the  songs  of  the  troubadours, 
and  the  baudekht,3  the  canopy  of  Bagdad,4  became  com- 
mon in  the  churches  of  Italy.  In  Sicily  and  in  Venice 
the  textile  industries  of  the  East  found  place,  and  made 
their  way  even  to  the  Scandinavian  peninsula.5 

We  therefore  have  this  state  of  affairs :  There  was 
abundant  intercourse  between  the  East  and  West  for 

1  J.  R.  Green,  Short  History  of  the  English  People,  New  York,  1890, 
p.  66. 

2  W.  Besant,  London,  New  York,  1892,  p.  43. 

3  Baldakin,  baldekin,  baldachino. 

4  Italian  Baldaeco. 

5  J.  K.  Mumford,  Oriental  Rugs,  New  York,  1901,  p.  18. 


110  THE  HINDU-ARABIC  NUMERALS 

some  centuries  before  the  Hindu  numerals  appear  in 
any  manuscripts  in  Christian  Europe.  The  numerals 
must  of  necessity  have  been  known  to  many  traders  in 
a  country  like  Italy  at  least  as  early  as  the  ninth  century, 
and  probably  even  earlier,  but  there  was  no  reason  for 
preserving  them  in  treatises.  Therefore  when  a  man  like 
Gerbert  made  them  known  to  the  scholarly  circles,  he 
was  merely  describing  what  had  been  familiar  in  a  small 
way  to  many  people  in  a  different  walk  of  life. 

Since  Gerbert 1  was  for  a  long  time  thought  to  have 
been  the  one  to  introduce  the  numerals  into  Italy,2  a 
brief  sketch  of  this  unique  character  is  proper.  Born  of 
humble  parents,3  this  remarkable  man  became  the  coun- 
selor and  companion  of  kings,  and  finally  wore  the  papal 
tiara  as  Sylvester  II,  from  999  until  his  death  in  1003.4 
He  was  early  brought  under  the  influence  of  the  monks 
at  Aurillac,  and  particularly  of  Kaimund,  who  had  been 
a  pupil  of  Odo  of  Cluny,  and  there  in  due  time  he  him- 
self took  holy  orders.  He  visited  Spain  in  about  967  in 
company  with  Count  Borel,5  remaining  there  three  years, 

1  Or  Girbert,  the  Latin  forms  Gerbertus  and  Girbertus  appearing 
indifferently  in  the  documents  of  his  time. 

2  See,  for  example,  J.  C.  Heilbronner,  Ilistoria  matheseos  universal, 
p.  740. 

3  "  Obscuro  loco  natum,"  as  an  old  chronicle  of  Aurillac  has  it. 

4  N.  Bubnov,  Gerberti  posted  Silvesiri  II  pupae  opera  mathematica, 
Berlin,  1899,  is  the  most  complete  and  reliable  source  of  information  ; 
Picavet,  loc.  cit.,  Gerbert  etc.;  Olleris,  (Euvres  de  Gerbert,  Paris,  18(57  ; 
Havet,  Lettresde  Gerbi  it.  Paris.  1889;  II.  Weissenborn,  Gerbert;  Bei- 
triit/c  zur  Kenntnis  </</■  Mathemalik  des  Mittdnltirs.  Berlin,  1888,  and 
Zur  Geschichte  der  Einfuhrung  der  jetzigen  Ziffem  in  Europa  durch 
Gerbert,  Berlin,  1892;  Biidinger,  Ucber  Gerberts  urissenschctftlieke  und 
politische  Stellung,  Cassel,  1851;  Richer,  "  Historiarum  liber  III,"  in 
Bubnov,  loc.  cit.,  pp.  37(5-381  ;  Nagl,  Gerbert  und  die  Bechenkunst  des 
10.  Jahrhunderts,  Vienna,  1888. 

c  Richer  tells  of  the  visit,  to  Aurillac  by  Borel,  a  Spanish  noble- 
man, just  as  Gerbert  was  entering  into  young  manhood.    He  relates 


DEFINITE  INTRODUCTION  INTO  EUROPE     111 

and  studying  under  Bishop  Hatto  of  Vich,1  a  city  in  the 
province  of  Barcelona, 2  then  entirely  under  Christian 
rule.  Indeed,  all  of  Gerbert's  testimony  is  as  to  the  in- 
fluence of  the  Christian  civilization  upon  his  education. 
Thus  he  speaks  often  of  his  study  of  Boethius,3  so  that 
if  the  latter  knew  the  numerals  Gerbert  would  have 
learned  them  from  him.4  If  Gerbert  had  studied  in  any 
Moorish  schools  he  would,  under  the  decree  of  the  emir 
Hisham  (787-822),  have  been  obliged  to  know  Arabic, 
which  would  have  taken  most  of  his  three  years  in 
Spain,  and  of  which  study  we  have  not  the  slightest 
hint  in  any  of  his  letters.6  On  the  other  hand,  Barce- 
lona was  the  only  Christian  province  in  immediate  touch 
with  the  Moorish  civilization  at  that  time.6  Further- 
more we  know  that  earlier  in  the  same  century  King 
Alonzo  of  Asturias  (d.  910)  confided  the  education  of 
his  son  Ordono  to  the  Arab  scholars  of  the  court  of  the 

how  affectionately  the  abbot  received  him,  asking  if  there  were  men  in 
Spain  well  versed  in  the  arts.  Upon  Borel's  reply  in  the  affirmative, 
the  abbot  asked  that  one  of  his  young  men  might  accompany  him  upon 
his  return,  that  he  might  carry  on  his  studies  there. 

1  Vicus  Ausona.    Hatto  also  appears  as  Atton  and  Hatton. 

2  This  is  all  that  we  know  of  his  sojourn  in  Spain,  and  this  comes 
from  his  pupil  Richer.  The  stories  told  by  Adhemar  of  Chabanois,  an 
apparently  ignorant  and  certainly  untrustworthy  contemporary,  of  his 
going  to  Cordova,  are  unsupported.  (See  e.g.  Picavet,  p.  34.)  Never- 
theless this  testimony  is  still  accepted:  K.  von  Raumer,  for  example 
(Geschichte  der  Piklagogik,  6th  ed.,  1890,  Vol.  I,  p.  6),  says  "Mathe- 
matik  studierte  man  im  Mittelalter  bei  den  Arabern  in  Spanien.  Zu 
ihnen  gieng  Gerbert,  nachmaliger  Pabst  Sylvester  II." 

3  Thus  in  a  letter  to  Aldaberon  he  says :  "  Quos  post  repperimus 
speretis,  id  est  VIII  volumina  Boeti  de  astrologia.  praeclarissima  quoque 
flgurarum  geometric,  aliaque  non  minus  admiranda"  (Epist.  8).  Also 
in  a  letter  to  Rainard  (Epist.  130),  he  says:  "Ex  tuis  sumptibus  fac 
ut  michi  scribantur  M.  Manlius  (Manilius  in  one  MS.)  de  astrologia." 

4  Picavet,  loc.  cit.,  p.  31. 

5  Picavet,  loc.  cit.,  p.  36. 

6  Havet,  loc.  cit.,  p.  vii. 


112  THE  HINDU-ARABIC  NUMERALS 

wall  of  Saragossa,1  so  that  there  was  more  or  less  of 
friendly  relation  between  Christian  and  Moor. 

After  his  three  years  in  Spain,  Gerbert  went  to  Italy, 
about  970,  where  he  met  Pope  John  XIII,  being  by  him 
presented  to  the  emperor  Otto  I.  Two  years  later  (972), 
at  the  emperor's  request,  he  went  to  Rheims,  where  he 
studied  philosophy,  assisting  to  make  of  that  place  an  ed- 
ucational center ;  and  in  983  he  became  abbot  at  Bobbio. 
The  next  year  he  returned  to  Rheims,  and  became  arch- 
bishop of  that  diocese  in  991.  For  political  reasons  he 
returned  to  Italy  in  996,  became  archbishop  of  Ravenna 
in  998,  and  the  following  year  was  elected  to  the  papal 
chair.  Far  ahead  of  his  age  in  wisdom,  he  suffered  as 
many  such  scholars  have  even  in  times  not  so  remote 
by  being  accused  of  heresy  and  witchcraft.  As  late  as 
1522,  in  a  biography  published  at  Venice,  it  is  related 
that  by  black  art  he  attained  the  papacy,  after  having 
given  his  soul  to  the  devil.2  Gerbert  was,  however, 
interested  in  astrology,3  although  this  was  merely  the 
astronomy  of  that  time  and  was  such  a  science  as  any 
learned  man  would  wish  to  know,  even  as  to-day  we  wish 
to  be  reasonably  familiar  with  physics  and  chemistry. 

That  Gerbert  and  his  pupils  knew  the  gobar  numer- 
als is  a  fact  no  longer  open  to  controversy.4  Berneli- 
nus  and  Richer5  call  them  by  the  well-known  name  of 

1  Picavet,  loc.  cit.,  p.  37. 

2  "  Con  sinistra  arti  conseguri  la  dignita  del  Pontificate  .  .  .  La- 
sciato  poi  1'  abito,  e'l  monasterio,  e  datosi  tutto  in  potere  del  diavolo." 
[Quoted  in  Bombelli,  Vantica  numerazione Italica,  Rome,  1876,  p.  41  n,] 

8  He  writes  from  Rheims  in  984  to  one  Lupitus,  in  Barcelona,  say- 
ing: "Itaque  librum  de  astrologia  translatum  a  te  niichi  petenti  di- 
rige,"  presumably  referring  to  some  Arabic  treatise.  [Epist.  no.  24 
of  the  Havet  collection,  p.  19.]  4  See  Bubnov,  loc.  cit.,  p.  x. 

5  011eris,loc.  cit.,  p.  361,1. 15,forBernelinus;  and  Bubnov,  loc.  cit., 
p.  381,  1.4,  for  Richer. 


DEFINITE  INTRODUCTION  INTO  EUROPE     113 

"  caracteres,"  a  word  used  by  Radulph  of  Laon  in  the 
same  sense  a  century  later.1  It  is  probable  that  Gerbert 
was  the  first  to  describe  these  gobar  numerals  in  any 
scientific  way  in  Christian  Europe,  but  without  the  zero.j 
If  he  knew  the  latter  he  certainly  did  not  understand 
its  use.2 

The  question  still  to  be  settled  is  as  to  where  he 
found  these  numerals.  That  he  did  not  bring  them  from 
Spain  is  the  opinion  of  a  number  of  careful  investiga- 
tors.3 This  is  thought  to  be  the  more  probable  because 
most  of  the  men  who  made  Spain  famous  for  learning 
lived  after  Gerbert  was  there.  Such  were  Ibn  Sina  / 
(Avicenna)  who  lived  at  the  beginning,  and  Gerber  of 
Seville  who  flourished  in  the  middle,  of  the  eleventh 
century,  and  Abu  Roshd  (Averroes)  who  lived  at  the 
end  of  the  twelfth.4    Others  hold  that  his  proximity  to 

1  Woepcke  found  this  in  a  Paris  MS.  of  Radulph  of  Laon,  c.  1100. 
[Propagation,  p.  240.]  "  Et  prima  quidem  trium  spaciorum  superductio 
unitatis  caractere  inscribitur,  qui  chaldeo  nomine  dicitur  igin."  See 
also  Alfred  Nagl,  "  Der  arithmetische  Tractat  des  Radulph  von 
Laon"  (Abhandlungen  zur  Geschichte  der  Mathematik,  Vol.V, pp.  85- 
133),  p.  97. 

2  Weissenborn,  loc.  cit.,  p.  239.  When  Olleris  (CEuvres  de  Gerbert, 
Paris,  1807,  p.  cci)  says,  "  C'est  a  lui  et  non  point  aux  Arabes,  que 
PEurope  doit  son  systeme  et  ses  signes  de  numeration,"  he  exaggerates, 
since  the  evidence  is  all  against  his  knowing  the  place  value.  Friedlein 
emphasizes  this  in  the  Zeitschrift  fiir  Mathematik  und  Physik,  Vol.  XII 
(1807),  Literaturzeitung,  p.  70:  "  Fiir  das  System  unserer  Numeration 
ist  die  Null  das  wesentlichste  Merkmal,  und  diese  kannte  Gerbert  nicht. 
Er  selbst  schrieb  alle  Zahlen  mit  den  romischen  Zahlzeichen  und  man 
kann  ihm  also  nicht  verdanken,  was  er  selbst  nicht  kannte." 

8  E.g.,  Chasles,  Biidinger,  Gerhardt,  and  Richer.  So  Martin  (Re- 
cherches  nouvelles  etc.)  believes  that  Gerbert  received  them  from  Boe- 
thius  or  his  followers.   See  Woepcke,  Propagation,  p.  41. 

4  Biidinger,  loc.  cit.,  p.  10.  Nevertheless,  in  Gerbert's  time  one  Al- 
Mansur,  governing  Spain  under  the  name  of  Hisham  (970-1002),  called 
from  the  Orient  Al-Begani  to  teach  his  son,  so  that  scholars  were 
recognized.    [Picavet,  p.  30.] 


114  THE  HINDU-ARABIC  NUMERALS 

the  Arabs  for  three  years  makes  it  probable  that  he  as- 
similated some  of  their  learning,  in  spite  of  the  fact 
that  the  lines  between  Christian  and  Moor  at  that  time 
were  sharply  drawn.1  Writers  fail,  however,  to  recog- 
nize that  a  commercial  numeral  system  would  have 
been  more  likely  to  be  made  known  by  merchants  than 
by  scholars.  The  itinerant  peddler  knew  no  forbidden 
pale  in  Spain,  any  more  than  he  lias  known  one  in  other 
lands.  If  the  gobar  numerals  were  used  for  marking 
wares  or  keeping  simple  accounts,  it  was  he  who  would 
have  known  them,  and  who  would  have  been  the  one 
rather  than  any  Arab  scholar  to  bring  them  to  the  in- 
quiring mind  of  the  young  French  monk.  The  facts 
that  Gerbert  knew  them  only  imperfectly,  that  lie  used 
them  solely  for  calculations,  and  that  the  forms  are  evi- 
dently like  the  Spanish  gobar,  make  it  all  the  more 
probable  that  it  was  through  the  small  tradesman  of  the 
Moors  that  this  versatile  scholar  derived  his  knowledge. 
Moreover  the  part  of  the  geometry  bearing  his  name,  and 
that  seems  unquestionably  his,  shows  the  Arab  influence7 
proving  that  he  at  least  came  into  contact  with  the 
transplanted  Oriental  learning,  even  though  imperfectly.2 
There  was  also  the  persistent  Jewish  merchant  trading 
with  both  peoples  then  as  now,  always  alive  to  the  ac- 
quiring of  useful  knowledge,  and  it  would  be  very  natu- 
ral for  a  man  like  Gerbert  to  welcome  learning  from 
such  a  source. 

On  the  other  hand,  the  two  leading  sources  of  infor- 
mation as  to  the  life  of  Gerbert  reveal  practically  noth- 
ing_to  show  that  he  came  within  the  Moorish  sphere  of 
influence  during  his  sojourn   in   Spain.     These  sources 

1  Weissenborn,  loc.  cit.,  p.  235.  2  Ibid.,  p.  234. 


DEFINITE  INTRODUCTION  INTO  EUROPE     115 

are  his  letters  and  the  history  written  by  Richer.  Gerbert 
was  a  master  of  the  epistolary  art,  and  his  exalted  posi- 
tion led  to  the  preservation  of  his  letters  to  a  degree 
that  would  not  have  been  vouchsafed  even  by  their 
classic  excellence.1  Richer  was  a  monk  at  St.  Remi  de 
Rheims,  and  was  doubtless  a  pupil  of  Gerbert.  The  lat- 
ter, when  archbishop  of  Rheims,  asked  Richer  to  write  a 
history  of  his  times,  and  this  was  done.  The  work  lay 
in  manuscript,  entirely  forgotten  until  Pertz  discovered 
it  at  Bamberg  in  1833.2  The  work  is  dedicated  to  Ger- 
bert as  archbishop  of  Rheims,3  and  would  assuredly  have 
testified  to  such  efforts  as  he  may  have  made  to  secure 
the  learning  of  the  Moors. 

Now  it  is  a  fact  that  neither  the  letters  nor  this  his- 
tory makes  any  statement  as  to  Gerbert's  contact  with 
the  Saracens.  The  letters  do  not  speak  of  the  Moors, 
of  the  Arab  numerals,  nor  of  Cordova.  Spam  is  not 
referred  to  by  that  name,  and  only  one  Spanish -scholar 
is  mentioned.  In  one  of  his  letters  he  speaks  of  Joseph 
Ispanus,4  or  Joseph  Sapiens,  but  who  this  Joseph  the 
Wise  of  Spain  may  have  been  we  do  not  know.    Possibly 

1  These  letters,  of  the  period  983-997,  were  edited  by  Havet,  loc. 
cit.,  and,  less  completely,  by  Olleris,  loc.  cit.  Those  touching  mathe- 
matical topics  were  edited  by  Bubnov,  loc.  cit.,  pp.  98-106. 

2  He  published  it  in  the  Monumenta  Germaniae  historica,  "  Scrip- 
tores,"  Vol.  Ill,  and  at  least  three  other  editions  have  since  ap- 
peared, viz.  those  by  Guadet  in  1845,  by  roinsignon  in  1855,  and  by 
Waitz  in  1877. 

3  Domino  ac  beatissimo  Patri  Gerberto,  Remorum  archiepiscopo, 
Richerus  Monchus,  Gallorum  congressibus  in  volumine  regerendis, 
imperii  tui,  pater  sanctissime  Gerberte,  auctoritas  seminarium  dedit. 

4  In  epistle  17  (Havet  collection)  he  speaks  of  the  "  De  multipli- 
catione  et  divisione  numerorum  libellum  a  Joseph  Ispano  editum  abbas 
Warnerius"  (a  person  otherwise  unknown).  In  epistle  25  he  says: 
"De  multiplicatione  et  divisione  numerorum,  Joseph  Sapiens  sen- 
tentias  quasdain  edidit." 


116  THE  HINDU-ARABIC  NUMERALS 

it  was  he  who  contributed  the  morsel  of  knowledge  so 
imperfectly  assimilated  by  the  young  French  monk.1 
Within  a  few  years  after  Gerbert's  visit  two  young  Span- 
ish monks  of  lesser  fame,  and  doubtless  with  not  that 
keen  interest  in  mathematical  matters  which  Gerbert  had, 
regarded  the  apparently  slight  knowledge  which  they  had 
of  the  Hindu  numeral  forms  as  worthy  of  somewhat  per- 
manent record2  in  manuscripts  which  they  were  transcrib- 
ing. The  fact  that  such  knowledge  had  penetrated  to  their 
modest  cloisters  in  northern  Spam  —  the  one  Albelda  or 
Albaida  —  indicates  that  it  was  rather  widely  diffused. 

Gerbert's  treatise  Libellus  de  numerorum  divisione  3  is 
characterized  by  Chasles  as  "  one  of  the  most  obscure 
documents  in  the  history  of  science."  4  The  most  com- 
plete information  in  regard  to  this  and  the  other  mathe- 
matical works  of  Gerbert  is  given  by  Bubnov,5  who 
considers  this  work  to  be  genuine.6 

1  H.  Suter,  "  Zur  Frage  liber  den  Josephus  Sapiens,"  Bibliotheca 
Mathematical  Vol.  VIII  (2),  p.  84  ;  Weissenborn,  Einfiihruny,  p.  14  ; 
also  his  Gerbert;  M.  Steinschneider,  in  Bibliotheca  Malhematica,  1893, 
p.  68.  Wallis  (Algebra,  1685,  chap.  14)  went  over  the  list  of  Spanish 
Josephs  very  carefully,  but  could  find  nothing  save  that  "Josephus 
Hispanus  seu  Josephus  sapiens  videtur  aut  Maurus  fuisse  aut  alius 
quia  in  Hispania." 

2  P.  Ewald,  Mittheilungen,  Neues  Archiv  d.  Gesellschaft  fiir  iiltere 
deidsche  Geschichtskunde,  Vol.  VIII,  1883,  pp.  354-364.  One  of  the 
manuscripts  is  of  976  a. i>.  and  the  other  of  992  a. d.  See  also  Franz 
Steffens,  Lateinische  Paldographie,  Freiburg  (Sclrweiz),  1903,  pp. 
xxxix-xl.    The  forms  are  reproduced  in  the  plate  on  page  140. 

3  It  is  entitled  Constantino  suo  Gerbertus  scolasticus,  because  it  was 
addressed  to  Constantine,  a  monk  of  the  Abbey  of  Fleury.  The  text 
of  the  letter  to  Constantine,  preceding  the  treatise  on  the  Abacus,  is 
given  in  the  Comptes  rendus,  Vol.  XVI  (1843),  p.  295.  This  book  seems 
to  have  been  written  c.  980  a. i>.    [Bubnov,  loc.  cit.,  p.  6.] 

4  "  Histoire  de  l'Arithm&ique,"  Comptes  rendus,  Vol.  XVI  (1843), 
pp.  156,  281.  s  Loc,  cjt-)  Gerberti  Opera  etc. 

G  Friedlein  thought  it  spurious.  See  Zeitschrift  fur  Mathematik  und 
Physik,  Vol.  XII  (1867),  Hist.-lit.  suppl.,  p.  74.    It  was  discovered  in 


DEFINITE  INTRODUCTION  INTO  EUROPE     117 

So  little  did  Gerbert  appreciate  these  numerals  that 
in  his  works  known  as  the  Regula  de  abaco  computi  and 
the  Libellus  he  makes  no  use  of  them  at  all,  employing 
only  the  Roman  forms.1  Nevertheless  Bernelinus  2  refers 
to  the  nine  gobar  characters.3  These  Gerbert  had  marked 
on  a  thousand  jeto7is  or  counters,4  using  the  latter  on  an 
abacus  which  he  had  a  sign-maker  prepare  for  him.5 
Instead  of  putting  eight  counters  in  say  the  tens'  column, 
Gerbert  would  put  a  single  counter  marked  8,  and  so 
for  the  other  places,  leaving  the  column  empty  where 
we  would  place  a  zero,  but  where  he,  lacking  the  zero, 
had  no  counter  to  place.  These  counters  he  possibly 
called  caracteres,  a  name  which  adhered  also  to  the  fig- 
ures themselves.  It  is  an  interesting  speculation  to  con- 
sider whether  these  apices,  as  they  are  called  in  the 
Boethius  interpolations,  were  in  any  way  suggested  by 
those  Roman  jetons  generally  known  in  numismatics 
as  tesserae,  and  bearing  the  figures  I-XVI,  the  sixteen 
referring  to   the  number   of    assi  in  a  sestertius.6   The 

the  library  of  the  Benedictine  monastry  of  St.  Peter,  at  Salzburg,  and 
was  published  by  Peter  Bernhard  Pez  in  1721.  Doubt  was  first  cast 
upon  it  in  the  Olleris  edition  (CEuvres  de  Gerbert).  See  Weissenborn, 
Gerbert,  pp.  2,  6,  168,  and  Picavet,  p.  81.  Hock,  Cantor,  and  Th.  Martin 
place  the  composition  of  the  work  at  c.  990  when  Gerbert  was  in  Ger- 
many, while  Olleris  and  Picavet  refer  it  to  the  period  when  he  was  at 
Rheims. 

1  Picavet,  loc.  cit.,  p.  182. 

2  Who  wrote  after  Gerbert  became  pope,  for  he  uses,  in  his  preface, 
the  words,  "a  domino  pape  Gerberto."  He  was  quite  certainly  not 
later  than  the  eleventh  century ;  we  do  not  have  exact  information 
about  the  time  in  which  he  lived. 

3  Picavet,  loc.  cit.,  p.  182.  Weissenborn,  Gerbert,  p.  227.  In  Olleris, 
Liber  Abaci  (of  Bernelinus),  p.  361. 

4  Richer,  in  Bubnov,  loc.  cit.,  p.  381. 

5  Weissenborn,  Gerbert,  p.  241. 

6  Writers  on  numismatics  are  quite  uncertain  as  to  their  use.  See 
F.  Gnecchi,  Monete  liomane,  2d  ed.,  Milan,  1900,  cap.  XXXVII.   For 


118  THE  HINDU-ARABIC  NUMERALS 

name  apices  adhered  to  the  Hindu-Arabic  numerals  until 
the  sixteenth  century.1 

To  the  figures  on  the  apices  were  given  the  names 
Igm,  andras,  ormis,  arbas,  quimas,  calctis  or  caltis,  zenis, 
temenias,  celentis,  sipos,2  the  origin  and  meaning  of 
which  still  remain  a  mystery.  The  Semitic  origin  of 
several  of  the  words  seems  probable.     Wahud,  thaneine, 

pictures  of  old  Greek  tesserae  of  Sarmatia,  see  S.  Ambrosoli,  Monete 
Greche,  Milan,  1899,  p.  202. 

1  Thus  Tzwivel's  arithmetic  of  1507,  fol.  2,  v.,  speaks  of  the  ten  fig- 
ures as  "  characteres  sive  numerorum  apices  a  diuo  Seuerino  Boetio." 

2  Weissenborn  uses  sipos  for  0.  It  is  not  given  by  Bernelinus,  and 
appears  in  Radulph  of  Laon,  in  the  twelfth  century.  See  Gunther\s 
Geschichte,  p.  98,  n.  ;  Weissenborn,  p.  11  ;  Pihan,  Expost  etc.,  pp. 
xvi-xxii. 

In  Friedlein's  Boetius,  p.  396,  the  plate  shows  that  all  of  the  six  im- 
portant manuscripts  from  which  the  illustrations  are  taken  contain  the 
symbol,  while  four  out  of  five  which  give  the  words  use  the  word  sipos 
for  0.  The  names  appear  in  a  twelfth-century  anonymous  manuscript 
in  the  Vatican,  in  a  passage  beginning 

Ordine  primigeno  sibi  nomen  possidet  igin. 
Andras  ecce  locum  mox  uendicat  ipse  secundum 
Ormis  post  numeros  incompositus  sibi  primus. 

[Boncompagni  Bidletino,  XV,  p.  132.]  Turchill  (twelfth  century)  gives 
the  names  Igin,  andras,  hormis,  arbas,  (pumas,  caletis,  zenis,  temenias, 
celentis,  saying :  "  Has  autem  figuras,  ut  donnus  [dominus]  Gvillelmus 
Rx  testatur,  a  pytagoricis  habemus,  nomina  uero  ab  arabibus."  (Who 
the  William  R.  was  is  not  known.  Boncompagni  Bulletino  XV,  p.  130.) 
Radulph  of  Laon  (d.  1131)  asserted  that  they  were  Chaldean  {Propa- 
gation, p.  48  n.).  A  discussion  of  the  whole  cpiestion  is  also  given  in 
E.  C.  Bayley,  loc.  cit.  Huet,  writing  in  1079,  asserted  that  they  were 
of  Semitic  origin,  as  did  Nesselmann  in  spite  of  his  despair  over  ormis, 
calctis,  and  celentis ;  see  Woepcke>  Propagation,  p.  48.  The  names 
were  used  as  late  as  the  fifteenth  century,  without  the  zero,  but  with 
the  superscript  dot  for  10's,  two  dots  for  100's,  etc.,  as  among  the 
early  Arabs.  Gerhardt  mentions  having  seen  a  fourteenth  or  fifteenth 
century  manuscript  in  the  Bibliotheca  Amploniana  with  the  names 
"  Ingnin,  andras,  arniis,  arbas,  quinas,  calctis,  zencis,  zemenias,  zcelen- 
tis,"  and  the  statement  "  Si  umim  punctum  super  ingnin  ponitur,  X 
significat.  ...  Si  duo  puncta  super  .  .  .  figuras  superponunter,  fiet 
decuplim  illius  quod  cum  uno  puncto  significabatur,"  in  Monats- 
bericMe  der  K.  P.  Akad.  d.  Wiss.,  Berlin,  1807,  p.  40. 


DEFINITE  INTRODUCTION  INTO  EUROPE     119 

thalata,  arba,  kumsa,  setta,  sebba,  timinia,  taseud  are  given 
by  the  Rev.  R.  Patrick  1  as  the  names,  in  an  Arabic  dia- 
lect used  in  Morocco,  for  the  numerals  from  one  to  nine. 
Of  these  the  words  for  four,  five,  and  eight  are  strikingly 
like  those  given  above. 

The  name  apices  was  not,  however,  a  common  one  in 
later  times.  Notae  was  more  often  used,  and  it  finally 
gave  the  name  to  notation.2  Still  more  common  were 
the  names  figures,  ciphers,  signs,  elements,  and  characters.3 

So  little  effect  did  the  teachings  of  Gerbert  have  in 
making  known  the  new  numerals,  that  O'Creat,  who 
lived  a  century  later,  a  friend  and  pupil  of  Adelhard 

1  A  chart  of  ten  numerals  in  200  tongues,  by  Rev.  R.  Patrick,  Lon- 
don, 1812. 

2  "Numeratio  figuralis  est  cuiusuis  numeri  per  notas,  et  figuras 
numerates  descriptio."  [Clichtoveus,  edition  of  c.  1507,  fol.  C  ii,  v.] 
"  Aristoteles  enim  uoces  rerum  (rvupoXa  uocat :  id  translation,  sonat 
notas."  [Noviomagus,  Be  Numeris  Libri  II,  cap.  vi.]  "  Alphabetum 
decern  notarum."  [Schonerus,  notes  to  Ramus,  1586,  p.  3  seq.]  Richer 
says:  "novemnumero  notas  omnemnumerumsignificantes."  [Bubnov, 
loc.  cit.,  p.  381.] 

3  "  II  y  a  dix  Characteres,  autrement  Figures,  Notes,  ou  Elements." 
[Peletier,  edition  of  1607,  p.  13.]  "  Numerorum  notas  alij  figuras,  alij 
signa,  alij  characteres  uocant."  [Glareanus,  1545  edition,  f.  9,  r.] 
"  Per  figuras  (quas  zyphras  uocant)  assignationem,  quales  sunt  lire 
notulse,  1.  2.  3.  4.  .  .  ."  [Noviomagus,  De  Numeris  Libri  II,  cap.  vi.] 
Gemma  Frisius  also  uses  elementa  and  Cardan  uses  literae.  In  the  first 
arithmetic  by  an  American  (Greenwood,  1729)  the  author  speaks  of 
"a  few  Arabian  Charecters  or  Numeral  Figures,  called  Digits'1''  (p.  1), 
and  as  late  as  1790,  in  the  third  edition  of  J.  J.  Blassiere's  arithmetic 
(1st  ed.  1769),  the  name  characters  is  still  in  use,  both  for  "  de  Latynsche 
en  de  Arabische  "  (p.  4),  as  is  also  the  term  "  Cyfferletters  "  (p.  6,  n.). 
Ziffer,  the  modern  German  form  of  cipher,  was  commonly  used  to 
designate  any  of  the  nine  figures,  as  by  Boeschenstein  and  Riese, 
although  others,  like  Kobel,  used  it  only  for  the  zero.  So  zifre  ap- 
pears in  the  arithmetic  by  Borgo,  1550  ed.  In  a  Munich  codex  of  the 
twelfth  century,  attributed  to  ( Jerland,  they  are  called  characters  only  : 
"Usque  ad  Villi,  enim  porrigitur  omnis  numerus  et  qui  supercrescit 
eisdem  designator  Karacteribus."  [Boncompagni  Bulletins,  Vol.  X. 
p.  607.] 


120  THE  HINDU-ARABIC  NUMERALS 

of  Bath,  used  the  zero  with  the  Roman  characters,  in 

contrast  to  Gerbert's  use  of  the  gobar  forms  without 

the  zero.1   O'Creat  uses  three  forms  for  zero,  o,  o,  and 

t,  as  in  Maximus  Planudes.    With  this  use  of  the  zero 

goes,  naturally,  a  place  value,  for  he  writes  III  III  for 

33,  ICCOO  and  I.  II.  r.  r  for  1200, 1. 0.  VIII.  IX  for  1089, 

and  I.  IIII.  IIII.  tttt  for  the  square  of  1200. 

The  period  from  the  time  of  Gerbert  until  after  the 

appearance   of   Leonardo's  monumental  work    may  be 

called  the  period  of  the  abacists.    Even  for  many  years 

after  the  appearance  early  in  the  twelfth  century  of  the 

books  explaining  the  Hindu  art  of  reckoning,  there  was 

strife  between  the  abacists,  the  advocates  of  the  abacus, 

and  the  algorists,  those  who  favored  the  new  numerals. 

The  words  cifra  and  algorismus  cifra  were  used  with 

a  somewhat  derisive  significance,  indicative  of  absolute 

uselessness,  as  indeed  the  zero  is  useless  on  an  abacus 

in  which  the  value  of  any  unit  is  given  by  the  column 

which  it  occupies.2   So  Gautier  de  Coincy  (1177-1236) 

hi  a  work  on  the  miracles  of  Mary  says: 

A  horned  beast,  a  sheep, 

An  algorismus-cipher, 

Is  a  priest,  who  on  such  a  feast  day 

Does  not  celebrate  the  holy  Mother.3 

So   the   abacus    held    the   field    for  a  long  time,  even 
against  the  new  algorism  employing  the  new  numerals. 

1  The  title  of  his  work  is  Prologus  N.  Ocreati  in  Helceph  (Arabic 
al-qeif,  investigation  or  memoir)  ad  Adelardum  Batensem  magistrum 
suum.  The  work  was  made  known  by  C.  Henry,  in  the  Zcitschriftfur 
Mathematik  und  Physik,  Vol.  XXV,  p.  129,  and  in  the  Abhandlungen 
zur  Geschichte  der  Mathematik,  Vol.  Ill ;  Weissenborn,  Gerbert,  p.  188. 

2  The  zero  is  indicated  by  a  vacant  column. 

8  Leo  Jordan,  loc.  cit.,  p.  170.  "Chifre  en  augorisme"  is  the  ex- 
pression used,  while  a  century  later  "giffre  en  argorisme  "  and  "  cyffres 
d'auyorisine"  are  similarly  used. 


DEFINITE  INTRODUCTION  INTO  EUROPE      121 

Geoffrey  Chaucer 1  describes  in  The  Miller's  Tale  the  clerk 

With         u  j Us  Almageste  and  bokes  grete  and  sniale, 
His  astrelabie,  longinge  for  his  art, 
His  augrim-stones  layen  faire  apart 
On  shelves  couched  at  his  beddes  heed." 

So,  too,  in  Chaucer's  explanation  of  the  astrolabe,2 
written  for  his  son  Lewis,  the  number  of  degrees  is  ex- 
pressed on  the  instrument  in  Hindu- Arabic  numerals: 
"  Over  the  whiche  degrees  ther  ben  noumbres  of  augrim, 
that  devyden  thilke  same  degrees  fro  fyve  to  fyve," 
and  "...  the  nombres  .  .  .  ben  writen  in  augrim," 
meaning  in  the  way  of  the  algorism.  Thomas  Usk 
about  1387  writes :  3  "a  sypher  in  augrim  have  no  might 
in  signification  of  it-selve,  yet  he  yeveth  power  hi  sig- 
nification to  other."  So  slow  and  so  painful  is  the  assimi- 
lation of  new  ideas. 

Bernelinus4  states  that  the  abacus  is  a  well-polished 
board  (or  table),  which  is  covered  with  blue  sand  and 
used  by  geometers  hi  drawing  geometrical  figures.  We 
have  previously  mentioned  the  fact  that  the  Hindus  also 
performed  mathematical  computations  in  the  sand,  al- 
though there  is  no  evidence  to  show  that  they  had  any 
column  abacus.5  For  the  purposes  of  computation, 
Bernelinus  continues,  the  board  is  divided  into  thirty 
vertical  columns,  tlnee  of  which  are  reserved  for  frac- 
tions.   Beginning  with  the  units  columns,  each  set  of 

i  The  Works  of  Geoffrey  Chaucer,  edited  by  W.  W.  Skeat,  Vol.  IV, 
Oxford,  1894,  p.  92. 

2  Loc.  cit.,  Vol.  Ill,  pp.  179  and  180. 

3  In  Book  II,  chap,  vii,  of  The  Testament  of  Love,  printed  with 
Chancers  Works,  loc.  cit.,  Vol.  VII,  London,  1897. 

4  Liber  Abacci,  published  in  Olleris,  CEuvres  de  Gerbert,  pp.  857-400. 

5  G.  R.  Kaye,  "The  Use  of  the  Abacus  in  Ancient  India,"  Journal 
and  Proceedings  of  the  Asiatic  Society  of  Bengal,  1908,  pp.  293-297. 


122  THE  HINDU-ARABIC  NUMERALS 

three  columns  (lineae  is  the  word  which  Bernelinus  uses) 
is  grouped  together  by  a  semicircular  arc  placed  above 
them,  while  a  smaller  arc  is  placed  over  the  units  col- 
umn and  another  joins  the  tens  and  hundreds  columns. 
Thus  arose  the  designation  arcus  pictagore  1  or  sometimes 
simply  areus.2  The  operations  of  addition,  subtraction, 
and  multiplication  upon  this  form  of  the  abacus  required 
little  explanation,  although  they  were  rather  extensively 
treated,  especially  the  multiplication  of  different  orders 
of  numbers.  But  the  operation  of  division  was  effected 
with  some  difficulty.  For  the  explanation  of  the  method 
of  division  by  the  use  of  the  complementary  difference,3 
long  the  stumbling-block  in  the  way  of  the  medieval 
arithmetician,  the  reader  is  referred  to  works  on  the  his- 
tory of  mathematics4  and  to  works  relating  particularly 
to  the  abacus.5 

Among  the  writers  on  the  subject  may  be  mentioned 
Abbo6  of  Fleury  (c.  970),  Heriger '  of  Lobbes  or  Laubach 

1  Liber  Abbaci,  by  Leonardo  Pisano,  loc.  cit.,  p.  1. 

2  Friedlein,  "Die  Entwickelung  des  Rechnensmit  Columnen,"  Zeit- 
schriftfur  Mathematik  und  Physik,  Vol.  X,  p.  247. 

3  The  divisor  6  or  16  being  increased  by  the  difference  4,  to  10  or 
20  respectively. 

4  E.g.  Cantor,  Vol.  I,  p.  882. 

5  Friedlein,  loc.  cit.;  Friedlein,  "  Cierbert's  Regeln  der  Division" 
anil  "Das  Rechnen  mit  Columnen  vor  deni  10.  Jahrhundert,"  Zeit- 
schrift  fur  Mathematik  und  Physik,  Vol.  IX  ;  Bubnov,  loc.  cit..  pp.  197- 
245;  M.  Chasles,  "Histoire  de  l'arithm6tique.  Kecherches  des  traces 
du  systeme  de  l'abacus,  apres  que  cette  niethode  a  gris  le  nom  d'Algo- 
risme.  —  Preuves  qu'a  tontes  les  epoques,  jusqu'au  xvie  siecle,  on  a  su 
que  l'arithm6tique  vulgaire  avait  pour  originecette  m^thode  ancienne," 
Comptes  rendus,  Vol.  XVII,  pp.  143-154,  also  "Regies  de  rabacus," 
Comptes  rendus,  Vol.  XVI,  pp.  218-246,  and  "Analyse  et  explication 
du  traite"  de  Gerbert,"  Comptes  rendus.  Vol.  XVI.  pp.  281-200. 

0  Bubnov,  loc  cit.,  pp.  203-204,  "  Abbonis  abacus." 
7  "Regulae  de  numerorum  abaci  rationibus,"  in  Bubnov,  loc.  cit., 
pp.  205-225. 


DEFINITE  INTRODUCTION  INTO  EUROPE     123 

(c.  950-1007),  and  Hermannus  Contractus1  (1013- 
1054),  all  of  whom  employed  only  the  Roman  numerals. 
Similarly  Adelhard  of  Bath  (c.  1130),  in  his  work  Regulae 
Abaci,2  gives  no  reference  to  the  new  numerals,  although  it 
is  certain  that  he  knew  them.  Other  writers  on  the  abacus 
who  used  some  form  of  Hindu  numerals  were  Gerland  3 
(first  half  of  twelfth  century)  and  Turchill4  (c.  1200). 
For  the  forms  used  at  this  period  the  reader  is  referred 
to  the  plate  on  page  88. 

After  Gerbert's  death,  little  by  little  the  scholars  of 
Europe  came  to  know  the  new  figures,  chiefly  through 
the  introduction  of  Arab  learning.  The  Dark  Ages  had 
passed,  although  arithmetic  did  not  find  another  advo- 
cate as  prominent  as  Gerbert  for  two  centuries.  Speak- 
ing of  this  great  revival,  Raoul  Glaber5  (985-c.  104(3),  a 
monk  of  the  great  Benedictine  abbey  of  Cluny,  of  the 
eleventh  century,  says :  "  It  was  as  though  the  world  had 
arisen  and  tossed  aside  the  worn-out  garments  of  ancient 
time,  and  wished  to  apparel  itself  in  a  white  robe  of 
churches."  And  with  this  activity  in  religion  came  a 
corresponding  interest  in  other  lines.  Algorisms  began 
to  appear,  and  knowledge  from  the  outside  world  found 

1  P.  Treutlein,  "Intorno  ad  alcuni  scritti  inediti  relativi  al  calcolo 
dell'  abaco,"  Bulletlno  di  bibliogrqfia  e  di  storia  delle  scienze  materna- 
tiche  efisiche,  Vol.  X,  pp.  589-647. 

2  "Intorno  ad  uno  scritto  inedito  di  Adelhardo  di  Bath  intitolato 
'Regulae  Abaci*'  "  B.  Boncompagni,  in  his  Bulletino,  Vol.  XIV, 
pp. 1-134. 

3  Treutlein,  loc.  cit. ;  Boncompagni,  "Intorno  al  Tractatus  de  Abaco 
di  Gerlando,"  Bulletino,  Vol.  X,  pp.  648-656. 

4  E.  Narducci,  "Intorno  a  due  trattati  inediti  d'abaco  contenuti 
in  due  codici  Vaticani  del  secolo  XII,"  Boncompagni  Bulletino,  Vol. 
XV,  pp.  111-162. 

6  See  Molinier,  Les  sources  de  Vhistoire  de  France,  Vol.  II,  Paris, 
1902,  pp.  2,  3. 


JC 


124  THE  HINDU-ARABIC  NUMERALS 

interested  listeners.  Another  Raoul,  or  Radulph,  to  whom 
we  have  referred  as  Radulph  of  Laon,1  a  teacher  in  the 
cloister  school  of  his  city,  and  the  brother  of  Anselm  of 
Laon2  the  celebrated  theologian,  wrote  a  treatise  on  music, 
extant  but  unpublished,  and  an  arithmetic  which  Nagl 
first  published  in  1890.3  The  latter  work,  preserved  to  us 
in  a  parchment  manuscript  of  seventy-seven  leaves,  con- 
tains a  curious  mixture  of  Roman  and  gobar  numerals,  the 
former  for  expressing  large  results,  the  latter  for  practical 
calculation.  These  gobar  "  caracteres  "  include  the  sipos 
(zero),  O,  of  which,  however,  Radulph  did  not  know 
the  full  significance;  showing  that  at  the  opening  of  the 
twelfth  century  the  system  was  still  uncertain  in  its  status 
in  the  church  schools  of  central  France. 

At  the  same  time  the  words  algorismus  and  cifra  were 
coming  into  general  use  even  in  non-mathematical  litera- 
ture. Jordan  4  cites  numerous  instances  of  such  use  from 
the  works  of  Alanus  ab  Insulis  5  (Alain  de  Lille),  Gau- 
tier  de  Coincy  (1177-1236),  and  others. 

Another  contributor  to  arithmetic  during  this  interest- 
ing period  was  a  prominent  Spanish  Jew  called  variously 
John  of  Luna,  John  of  Seville,  Johannes  Hispalensis, 
Johannes  Toletanus,  and  Johannes  Hispanensis  de  Luna.6 

1  Cantor,  Geschichte,  Vol.  I,  p.  762.    A.  Nagl  in  the  Abhandlungen 
zur  Geschichte  der  Mathematik,  Vol.  V,  p.  85. 
2 1030-1117. 

3  Abhandlungen  zur  Geschichte  der  Mathematik,  Vol.  V,  pp.  85-133. 
The  work  begins  "Incipit  Liber  Radulfi  laudunensis  de  abaco." 

4  Materialien  zur  Geschichte  der  arabischen  Zahlzeichen  in  Frankreich, 
loc.  cit.  6  who  died  in  1202. 

6  Cantor,  Geschichte,Yo\.  1(3),  pp.  800-803  ;  Boncompagni,  Trattati, 
Tart  II.  M.  Steinschneider  ("Die  Mathematik  bei  den  Juden," 
BibliotJieca  Mathematical ol.  X  (2),  p.  79)  ingeniously  derives  another 
name  by  which  he  is  called  (Abendeuth)  from  Ibn  Daiid  (Son  of  David). 
Sec  also  Abhandlungen,  Vol.  HI,  p.  110. 


DEFINITE  INTRODUCTION  INTO  EUROPE     125 

His  date  is  rather  closely  fixed  by  the  fact  that  he  dedi- 
cated a  work  to  Raimund  who  was  archbishop  of  Toledo 
between  1130  and  1150. *  His  interests  were  chiefly  in 
the  translation  of  Arabic  works,  especially  such  as  bore 
upon  the  Aristotelian  philosophy.  From  the  standpoint 
of  arithmetic,  however,  the  chief  interest  centers  about  a 
manuscript  entitled  Joannis  Hispalensis  liber  Algorismi  de 
Practiea  Arismetrice  which  Boncompagni  found  in  what 
is  now  the  Bibliotheque  nationale  at  Paris.  Although  this 
distinctly  lays  claim  to  being  Al-Khowarazmfs  work,2 
the  evidence  is  altogether  against  the  statement,3  but  the 
book  is  quite  as  valuable,  since  it  represents  the  knowl- 
edge of  the  time  in  which  it  was  written.  It  relates  to  the 
operations  with  integers  and  sexagesimal  fractions,  in- 
cluding roots,  and  contains  no  applications.4 

Contemporary  with  John  of  Luna,  and  also  living  in 
Toledo,  was  Gherard  of  Cremona,5  who  has  sometimes 
been  identified,  but  erroneously,  with  Gernardus,6  the 

1  John  is  said  to  have  died  in  1157. 

2  For  it  says,  "  Ineipit  prologus  in  libro  alghoarismi  de  practiea 
arismetrice.  Qui  editus  est  a  magistro  Johanne  yspalensi."  It  is  pub- 
lished in  full  in  the  second  part  of  Boncompagni's  Trattati  d'aribmetica. 

3  Possibly,  indeed,  the  meaning  of  "libro  alghoarismi"  is  not  "to 
Al-Khowarazmi's  book,"  but  "to  a  book  of  algorism."  John  of  Luna 
says  of  it:  "Hoc  idem  est  illud  etiam  quod  .  .  .  alcorismus  dicere 
videtur."    [Trattati,  p.  68.] 

4  For  a  r6sume\  see  Cantor,  Vol.  I  (3),  pp.  800-803.  As  to  the  au- 
thor, see  Enestrom  in  the  Bibliotheca  Mathematical,  Vol.  VI  (3),  p.  114, 
and  Vol.  IX  (3),  p.  2. 

5  Born  at  Cremona  (although  some  have  asserted  at  Carmona,  in 
Andalusia)  in  1114;  died  at  Toledo  in  1187.  Cantor,  loc.  cit.;  Bon- 
compagni, Atti  d.  R.  Accad.  d.  n.  Lincei,  1851. 

6  See  Abhandlungen  zur  Geschichte  der  Mathematik,  Vol. XIV,  p.  149 ; 
Bibliotheca  Mathematica,  Vol.  IV  (3),  p.  206.  Boncompagni  had  a 
fourteenth-century  manuscript  of  his  work,  Gerardi  Cremonensis  artis 
vt(  trice  practice.  See  also  T.  L.  Heath,  The  Thirteen  Books  of  Euclid's 
Elements,  3  vols.,  Cambridge,  1908,  Vol.  I,  pp.  92-94  ;  A.  A.  Bjornbo, 


126  THE  HINDU-ARABIC  NUMERALS 

author  of  a  work  on  algorism.  He  was  a  physician,  an 
astronomer,  and  a  mathematician,  translating  from  the 
Arabic  both  in  Italy  and  in  Spain.  In  arithmetic  he  was 
influential  in  spreading  the  ideas  of  algorism. 

Four  Englishmen  —  Adelhard  of  Bath  (c.  1130),  Rob- 
ert of  Chester  (Robertus  Cestrensis,  c.  1143),  William 
Shelley,  and  Daniel  Morley  (1180)  —  are  known 1  to 
have  journeyed  to  Spain  in  the  twelfth  century  for  the 
purpose  of  studying  mathematics  and  Arabic.  Adelhard 
of  Bath  made  translations  from  Arabic  into  Latin  of  Al- 
Khowarazmfs  astronomical  tables2  and  of  Euclid's  Ele- 
ments,3 while  Robert  of  Chester  is  known  as  the  translator 
ijof  Al-Khowarazmf  s  algebra.4  There  is  no  reason  to  doubt 
that  all  of  these  men,  and  others,  were  familiar  with  the 
numerals  which  the  Arabs  were  using. 

The  earliest  trace  we  have  of  computation  with  Hindu 
numerals  in  Germany  is  in  an  Algorismus  of  1143,  now 
in  the  Hofbibliothek  in  Vienna.5    It  is  bound  in  with  a 

"Gerhard  von  Cremonas  Ubersetzung  von  Alkwarizmis  Algebra  und 
von  Euklids  Elementen,"  Bibliotheca  Mathematica,  Vol.  VI  (3),  pp. 
239-248.  i  Wallis,  Algebra,  1085,  p.  12  seq. 

2  Cantor,  Geschichte,  Vol.  1(3),  p.  906;  A.  A.  Bjornbo,  "Al-Chwa- 
rizmi's  trigonometriske  Tavler,"  Festskrifi  til  H.  G.  Zeuthen,  Copen- 
hagen, 1909,  pp.  1-17.  s  Heath,  loc.  cit.,  pp.  93-96. 

4  M.  Steinschneider,  Zeitschrift  der  deutschen  morgenldndischen  Ge- 
sellschaft,  Vol.  XXV,  1871,  p.  104,  and  Zeitschrift  fiir  Matheniatik  und 
Physik,  Vol.  XVI,  1871,  pp.  392-393;  M.  Curtze,  Centralblatt  fur 
Ilibliothekswesen,  1899,  p.  289;  E.  Wappler,  Zur  Geschichte  der  deut- 
schen Algebra  im  15.  Jahrhundert,  Programm,  Zwickau,  1887 ;  L.  C. 
Karpinski,  "Robert  of  Chester's  Translation  of  the  Algebra  of  Al- 
Khowarazmi,"  Bibliotheca  Mathematica,  Vol.  XI  (3),  p.  125.  He  is  also 
known  as  Robertus  Retinensis,  or  Roberl  of  Reading. 

6  Nagl,  A.,  "  Ueber  eine  Algorismus-Schrift  des  XII.  Jahrlmnderts 
und  iiber  die  Verbreitung  der  indisch-arabischen  Rechenkunst  und 
Zahlzeichen  im  christl.  Abendlande,"  in  the  Zeitschrift  fiir  Matheniatik 
und  Physik,  Hist. -lit.  Abth.,  Vol.  XXXIV,  p.  129.  Curtze,  Abhand- 
lungen  zur  Geschichte  der  Matheniatik,  Vol.  VIII,  pp.  1-27. 


DEFINITE  INTRODUCTION  INTO  EUROPE     127 

ComputuB  by  the  same  author  and  bearing  the  date  given. 
It  contains  chapters  "  De  additione,"  "  De  diminutione," 
"  De  mediatione,"  "  De  divisione,"  and  part  of  a  chap- 
ter on  multiplication.  The  numerals  are  in  the  usual  medi- 
eval forms  except  the  2,  which,  as  will  be  seen  from  the 
illustration,1  is  somewhat  different,  and  the  3,  which 
takes  the  peculiar  shape  h,  a  form  characteristic  of  the 
twelfth  century. 

It  was  about  the  same  time  that  the  Sefer  ha-Mispar,'2' 
the  Book  of  Number,  appeared  in  the  Hebrew  language. 
The  author,  Rabbi  Abraham  ibn  Meir  ibn  Ezra,3  was 
born  in  Toledo  (c.  1092).  In  1139  he  went  to  Egypt, 
Palestine,  and  the  Orient,  spending  also  some  years  in 
Italy.  Later  he  lived  in  southern  France  and  in  Eng- 
land. He  died  in  1167.  The  probability  is  that  he  ac- 
quired his  knowledge  of  the  Hindu  arithmetic 4  in  his 
native  town  of  Toledo,  but  it  is  also  likely  that  the 
knowledge  of  other  systems  which  he  acquired  on  travels 
increased  his  appreciation  of  this  one.  We  have  men- 
tioned the  fact  that  he  used  the  first  letters  of  the  Hebrew 
alphabet,  B  n  7  1  H  1  3  3  N,  for  the 
numerals  9  87G5  432  1,  and  a 
circle  for  the  zero.  The  quotation  in  the  note  given  be- 
low shows  that  he  knew  of  the  Hindu  origin  ;  but  in  his 
manuscript,  although  he  set  down  the  Hindu  forms,  he 
used  the  above  nine  Hebrew  letters  with  place  value  for 

4 

all  computations. 

1  See  line  a  in  the  plate  on  p.  148. 

2  Sefer  ha-Mispar,  Das  Buck  der  Zahl,  ein  hebrdisch-arithmetisehes 
Werk  des  R.  Abraham  ibn  Esra,  Moritz  Silberberg,  Frankfurt  a.M.,  1895. 

3  Browning's  "  Rabbi  ben  Ezra." 

4  ''Darum  haben  audi  die  Weisen  Indiens  all  ihre  Zahlen  durch 
neun  bezeichnet  und  Fornien  fiir  die  9  Ziffern  gebildet."  [Sefer  ha- 
Mispar,  loc.  cit.,  p.  2.1 


CHAPTER  VIII 
THE  SPREAD  OF  THE  NUMERALS  IN  EUROPE 

Of  all  the  medieval  writers,  probably  the  one  most  in- 
fluential hi  introducing  the  new  numerals  to  the  scholars 
of  Europe  was  Leonardo  Fibonacci,  of  Pisa.1  This  remark- 
able man,  the  most  noteworthy  mathematical  genius  of 
the  Middle  Ages,  was  born  at  Pis*a  about  11 75.2 

The  traveler  of  to-day  may  cross  the  Via  Fibonacci 
on  his  way  to  the  Campo  Santo,  and  there  he  may  see 
at  the  end  of  the  long  corridor,  across  the  quadrangle, 
the  statue  of  Leonardo  in  scholar's  garb.  Few  towns 
have  honored  a  mathematician  more,  and  few  mathema- 
ticians have  so  distinctly  honored  their  birthplace.  Leo- 
nardo was  born  in  the  golden  age  of  this  city,  the  period 
of  its  commercial,  religious,  and  intellectual  prosperity.3 

1  F.  Bonaini,  "Memoria  unica  sincrona  di  Leonardo  Fibonacci," 
Pisa,  1858,  republished*in  1867,  and  appearing  in  the  Giornale  Arca- 
dico,  Vol.  CXCVII  (N.  S.  MI);  Gaetano  Milanesi,  Bocumento  inedito  e 
sconosciuto  a  Lionardo  Fibonacci,  Roma,  18G7 ;  Guglielmini,  Elogio 
di  Lionardo  Pisano,  Bologna,  1812,  p.  35;  Libri,  Histoire  des  sci- 
ences matMmatiques,  Vol.  II,  p.  25;  D.  Martines,  Origine  e  progressi 
deW  aritmetica,  Messina,  1865,  p.  47 ;  Lucas,  in  Boncompagni  Bulle- 
tin*), Vol.  X,  pp.  129,239 ;  Besagne,  ibid.,  Vol.  IX,  p.  583;  Boncompagni, 
three  works  as  cited  in  Chap.  I;  G.  Enestrom,  "Ueber  zwei  angeb- 
liche  mathematische  Schulen  im  christlichen  Mittelalter,"  Bibliotheca 
Mathematica,  Vol.  VIII  (3),  pp.  252-262 ;  Boncompagni,  "Delia  vita 
e  delle  opere  di  Leonardo  Pisa  no,"  loc.  cit. 

2  The  date  is  purely  conjectural.  See  the  Bibliotheca  Mathematica, 
Vol.  IV  (3),  p.  215. 

8  An  old  chronicle  relates  that  in  1063  Pisa  fought  a  great  battle 
with  the  Saracens  at  Palermo,  capturing  six  ships,  one  being  "full  of 
wondrous  treasure,"  and  this  was  devoted  to  building  the  cathedral. 

128 


SPREAD  OF  THE  NUMERALS  IN  EUROPE   129 

Situated  practically  at  the  mouth  of  the  Arno,  Pisa 
formed  with  Genoa  and  Venice  the  trio  of  the  greatest 
commercial  centers  of  Italy  at  the  opening  of  the  thirteenth 
century.  Even  before  Venice  had  captured  the  Levan- 
tine trade,  Pisa  had  close  relations  with  the  East.  An 
old  Latin  chronicle  relates  that  in  1005  "Pisa  was  cap- 
tured by  the  Saracens,"  that  in  the  following  year  "  the 
Pisans  overthrew  the  Saracens  at  Reggio,"  and  that  in 
1012  "  the  Saracens  came  to  Pisa  and  destroyed  it."  The 
city  soon  recovered,  however,  sending  no  fewer  than  a 
hundred  and  twenty  ships  to  Syria  in  1099,1  founding 
a  merchant  colony  in  Constantinople  a  few  years  later,2 
and  meanwhile  carrying  on  an  interurban  warfare  in  Italy 
that  seemed  to  stimulate  it  to  great  activity.3  A  writer 
of  1114  tells  us  that  at  that  time  there  were  many  hea- 
then people  —  Turks,  Libyans,  Parthians,  and  Chalde- 
ans —  to  be  found  in  Pisa.  It  was  in  the  midst  of  such 
wars,  in  a  cosmopolitan  and  commercial  town,  in  a  cen- 
ter where  literary  work  was  not  appreciated,4  that  the 
genius  of  Leonardo  appears  as  one  of  the  surprises  of 
history,  warning  us  again  that  "we  should  draw  no 
horoscope ;  that  we  should  expect  little,  for  what  we 
expect  will  not  come  to  pass."  5 

Leonardo's  father  was  one  William,0  and  he  had  a 
brother    named    Bonaccingus,7    but    nothing    further   is 

i  Heyd,  loc.  cit.,  Vol.  I,  p.  149.  2  Ibid.,  p.  211. 

3  J.  A.  Symonds,  Renaissance  in  Italy.  The  Age  of  Despots.  New 
York,  1883,  p.  62.  4  Symonds,  loc.  cit.,  p.  79. 

5  J.  A.  Froude,  The  Science  of  History,  London,  1864.  "Un  brevet 
d'apothicaire  n'empecha  pas  Dante  d'etre  le  plus  grand  poete  de 
Tltalie,  et  ce  fut  un  petit  marchand  de  Pise  qui  donna  l'algebre  aux 
Chretiens."    [Libri,  Histoire,  Vol.  I,  p.  xvi.] 

6  A  document  of  1226,  found  and  published  in  1858,  reads:  "Leo- 
nardo bigollo  quondam  Guilielmi."        7  "Bonaccingo  germano  suo," 


130  THE  HINDU-ARABIC  NUMERALS 

known  of  his  family.  As  to  Fibonacci,  most  writers  1  have 
assumed  that  his  father's  name  was  Bonaccio,2  whence 
films  Bonaccii,  or  Fibonacci.  Others 3  believe  that  the 
name,  even  in  the  Latin  form  of  filius  Bonaccii  as  used 
in  Leonardo's  work,  was  simply  a  general  one,  like  our 
Johnson  or  Bronson  (Brown's  son)  ;  and  the  only  con- 
temporary evidence  that  we  have  bears  out  this  view. 
As  to  the  name  Bigollo,  used  by  Leonardo,  some  have 
thought  it  a  self -assumed  one  meaning  blockhead,  a  term 
that  had  been  applied  to  him  by  the  commercial  world 
or  possibly  by  the  university  circle,  and  taken  by  him 
that  he  might  prove  what  a  blockhead  could  do.  Mila- 
nesi,4  however,  has  shown  that  the  word  Bigollo  (or 
Pigollo)  was  used  in  Tuscany  to  mean  a  traveler,  and 
was  naturally  assumed  by  one  who  had  studied,  as  Leo- 
nardo had,  in  foreign  lands. 

Leonardo's  father  was  a  commercial  agent  at  Bugia, 
the  modern  Bougie,5  the  ancient  Saldae  on  the  coast  of 
Barbary,6  a  royal  capital  under  the  Vandals  and  again, 
a  century  before  Leonardo,  under  the  Beni  Hammad. 
It  had  one  of  the  best  harbors  on  the  coast,  sheltered  as 
it  is  by  Mt.  Lalla  Guraia,7  and  at  the  close  of  the  twelfth 
century  it  was  a  center  of  African  commerce.  It  was  here 
that  Leonardo  was  taken  as  a  child,  and  here  he  went  to 
school  to  a  Moorish  master.  When  he  reached  the  years 
of  young  manhood  he  started  on  a  tour  of  the  Medi- 
terranean Sea,  and  visited  Egypt,  Syria,  Greece,  Sicily, 
and  Provence,  meeting  with  scholars  as  well  as  with 

1  E.g.  Libri,  Guglielmini,  Tiraboschi.  2  Latin,  Bonaccius. 

3  Boncompagni  and  Milanesi.  4  Reprint,  p.  5. 

6  Whence  the  French  name  for  candle. 

6  Now  part  of  Algiers. 

7  E.  Reclus,  Africa,  New  York,  1893,  Vol.  II,  p.  253. 


SPREAD  OF  THE  NUMERALS  IN  EUROPE      131 

merchants,  and  imbibing  a  knowledge  of  the  various  sys- 
tems of  numbers  in  use  in  the  centers  of  trade.  All  these 
systems,  however,  he  says  he  counted  almost  as  errors 
compared  with  that  of  the  Hindus.1  Returning  to  Pisa, 
he  wrote  his  Liber  Abaci2  in  1202,  rewriting  it  in  1228.3 
In  this  work  the  numerals  are  explained  and  are  used 
in  the  usual  computations  of  business.  Such  a  treatise 
was  not  destined  to  be  popular,  however,  because  it 
was  too  advanced  for  the  mercantile  class,  and  too 
novel  for  the  conservative  university  circles.  Indeed,  at 
this  time  mathematics  had  only  slight  place  in  the  newly 
established  universities,  as  witness  the  oldest  known  stat- 
ute of  the  Sorbonne  at  Paris,  dated  1215,  where  the  sub- 
ject is  referred  to  only  in  an  incidental  way.4  The  period 
was  one  of  great  commercial  activity,  and  on  this  very 


1  "  Sed  hoc  totum  et  algorismum  atque  arcus  pictagore  quasi  erro- 
rem  computavi  respectu  modi  indorum."  Woepcke,  Propagation  etc., 
regards  this  as  referring  to  two  different  systems,  but  the  expression 
may  very  well  mean  algorism  as  performed  upon  the  Pythagorean 
arcs  (or  table). 

2  "  Book  of  the  Abacus,"1  this  term  then  being  used,  and  long  after- 
wards in  Italy,  to  mean  merely  the  arithmetic  of  computation. 

3  "  Incipit  liber  Abaci  a  Leonardo  filio  Bonacci  compositus  anno 
1202  et  c'orrectus  ab  eodem  anno  1228."  Three  MSS.  of  the  thirteenth 
century  are  known,  viz.  at  Milan,  at  Siena,  and  in  the  Vatican  library. 
The  work  wras  first  printed  by  Boncompagni  in  1857. 

4  I.e.  in  relation  to  the  quadrivium.  "Non  legant  in  festivis  diebus, 
nisi  Philosophos  et  rhetoricas  et  quadrivalia  et  barbarismum  et  ethi- 
cam,  si  placet."1  Suter,  Die  Mathematik  auf  den  Universitaten  des 
Mittelalters,  Zurich,  1887,  p.  56.  Roger  Bacon  gives  a  still  more 
gloomy  view  of  Oxford  in  his  time  in  his  Opus  minus,  in  the  Berum 
Britannicarum  medii  aevi  scriptores,  London,  1859,  Vol.  I,  p.  327.  For 
a  picture  of  Cambridge  at  this  time  consult  F.  W.  Newman,  The 
English  Universities,  translated  from  the  German  of  V.  A.  Iluber,  Lon- 
don, 1843,  Vol.1,  p.  61;  W.  W.  R.  Ball,  History  of  Mathematics  at 
Camtxridge,  1889;  S.  Gunther,  Geschichte  des  mathematischen  Unter- 
richts  im  deutschen  Mittelalter  bis  zum  Jahre  1525,  Berlin,  1887,  being 
Vol.  Ill  of  Monumenta  Germaniae paedagogica. 


132  THE  HINDU-ARABIC  NUMERALS 

account  such  a  book  would  attract  even  less  attention 
than  usual.1 

It  would  now  be  thought  that  the  western  world 
would  at  once  adopt  the  new  numerals  which  Leonardo 
had  made  known,  and  which  were  so  much  superior  to 
anything  that  had  been  in  use  in  Christian  Europe.  The 
antagonism  of  the  universities  would  avail  but  little,  it 
would  seem,  against  such  an  improvement.  It  must  be 
remembered,  however,  that  there  was  great  difficulty  in 
spreading  knowledge  at  this  time,  some  two  hundred  and 
fifty  years  before  printing  was  invented.  "Popes  and 
princes  and  even  great  religious  institutions  possessed  far 
fewer  books  than  many  farmers  of  the  present  age.  The 
library  belonging  to  the  Cathedral  Church  of  San  Mar- 
tino  at  Lucca  in  the  ninth  century  contained  only  nineteen 
volumes  of  abridgments  from  ecclesiastical  commenta- 
ries." 2  Indeed,  it  was  not  until  the  early  part  of  the  fif- 
teenth century  that  Palla  degli  Strozzi  took  steps  to  carry 
out  the  project  that  had  been  in  the  mind  of  Petrarch, 
the  founding  of  a  public  library.  It  was  largely  by 
word  of  mouth,  therefore,  that  this  early  knowledge  had 
to  be  transmitted.  Fortunately  the  presence  of- foreign 
students  in  Italy  at  this  time  made  this  transmission 
feasible.  (If  human  nature  was  the  same  then  as  now,  it 
is  not  impossible  that  the  very  opposition  of  the  faculties 
to  the  works  of  Leonardo  led  the  students  to  investigate 

1  On  the  commercial  activity  of  the  period,  it  is  known  that  bills 
of  exchange  passed  between  Messina  and  Constantinople  in  1161, 
and  that  a  bank  was  founded  at  Venice  in  1170,  the  Bank  of  San 
Marco  being  established  in  the  following  year.  The  activity  of  Bisa 
was  very  manifest  at  this  time.  Heyd,  loc.  cit.,  Vol.  II,  p.  5 ;  V.  Casa- 
grandi,  Storia  e  cronologia,  3d  ed.,  Milan,  1901,  p.  56. 

2  J.  A.  Symonds,  loc.  cit.,  Vol.  II,  p.  127. 


SPREAD  OF  THE  NUMERALS  IN  EUROPE      133 

them  the  more  zealously.)  At  Vicenza  in  1209,  for 
example,  there  were  Bohemians,  Poles,  Frenchmen, 
Burgundians,  Germans,  and  Spaniards,  not  to  speak  of 
representatives  of  divers  towns  of  Italy ;  and  what  was 
true  there  was  also  true  of  other  intellectual  centers. 
The  knowledge  could  not  fail  to  spread,  therefore,  and 
as  a  matter  of  fact  we  find  numerous  bits  of  evidence 
that  this  was  the  case.  Although  the  bankers  of  Flor- 
ence were  forbidden  to  use  these  numerals  in  1299,  and 
the  statutes  of  the  university  of  Padua  required  station- 
ers to  keep  the  price  lists  of  books  "  non  per  cifras,  sed 
per  literas  claros," 1  the  numerals  really  made  much 
headway  from  about  1275  on. 

It  was,  however,  rather  exceptional  for  the  common 
people  of  Germany  to  use  the  Arabic  numerals  before  the 
sixteenth  century,  a  good  witness  to  this  fact  being  the 
popular  almanacs.  Calendars  of  1457-1496  2  have  gener- 
ally the  Roman  numerals,  while  Kobel's  calendar  of  1518 
gives  the  Arabic  forms  as  subordinate  to  the  Roman. 
In  the  register  of  the  Kreuzschule  at  Dresden  the  Roman 
forms  were  used  even  until  1539. 

While  not  minimizing  the  importance  of  the  scientific 
work  of  Leonardo  of  Pisa,  we  may  note  that  the  more  pop- 
ular treatises  by  Alexander  de  Villa  Dei  (c.  1240  a.d.) 
and  John  of  Halifax  (Sacrobosco,  c.  1250  a.d.)  were 
much  more  widely  used,  and  doubtless  contributed  more 
to  the  spread  of  the  numerals  among  the  common  people. 

1  I.  Taylor,  The  Alphabet,  London,  1883,  Vol.  II,  p.  263. 

2  Cited  by  Unger's  History,  p.  15.  The  Arabic  numerals  appear  in 
a-Regensburg  chronicle  of  11(37  and  in  Silesia  in  1340.  See  Schmidt's 
EncyclojMdie  der  Erziehung,  Vol.  VI,  p.  726 ;  A.  Kuckuk,  "Die  Rechen- 
kunst  imsechzehnten  Jahrhundert,"  Festschrift  zur  dritten  Sacularfeier 
des  Berlinischen  Gymnasiums  zum  grauen  Elostcr,  Berlin,  1874,  p.  4. 


134  THE  HINDU-ARABIC  NUMERALS 

The  Carmen  de  Algorismo  1  of  Alexander  de  Villa  Dei 
was  written  in  verse,  as  indeed  were  many  other  text- 
books of  that  time.  That  it  was  widely  used  is  evidenced 
by  the  large  number  of  manuscripts  2  extant  in  European 
libraries.  Sacrobosco's  Algorismus?  in  which  some  lines 
from  the  Carmen  are  quoted,  enjoyed  a  wide  popularity 
as  a  textbook  for  university  instruction.4  The  work  was 
evidently  written  with  this  end  in  view,  as  numerous 
commentaries  by  university  lecturers  are  found.  Proba- 
bly the  most  widely  used  of  these  was  that  of  Petrus  de 
Dacia5  written  in  1291.  These  works  throw  an  interest- 
ing light  upon  the  method  of  instruction  in  mathematics 
in  use  hi  the  universities  from  the  thirteenth  even  to  the 
sixteenth  century.  Evidently  the  text  was  first  read  and 
copied  by  students.6  Following  this  came  line  by  line  an 
exposition  of  the  text,  such  as  is  given  in  Petrus  de 
Dacia's  commentary. 

Sacrobosco's  work  is  of  interest  also  because  it  was 
probably  due  to  the  extended  use  of  this  work  that  the 

1  The  text  is  given  in  Halliwell,  Rara  Mathematica,  London,  1839. 

2  Seven  are  given  in  Ashmole's  Catalogue  of  Manuscripts  in  the 
Oxford  Library,  1845. 

3  Maximilian  Curtze,  Petri  Philomeni  de  Dacia  in  Algorismum  Vid- 
garem  Johannis  de  Sacrobosco  commentarius,  una  cum  Algorismo  ipso, 
Copenhagen,  1897  ;  L.  C.  Karpinski,  "  Jordanus  Nemorarius  and  John 
of  Halifax,"  American  Mathematical  Monthly,  Vol.  XVII,  pp.  108-113. 

4  J.  Aschbacb,  Geschichte  der  Wiener  Universitat  im  ersten  Jahrhun- 
derte  Hires  Bestehens,  Wien,  1865,  p.  93. 

5  Curtze,  loc.  cit.,  gives  the  text. 

fi  Curtze,  loc.  cit.,  found  some  forty-five  copies  of  the  Algoris- 
mus  in  three  libraries  of  Munich,  Venice,  and  Erfurt  (Amploniana). 
Examination  of  two  manuscripts  from  the  Plimpton  collection  and 
the  Columbia  library  shows  such  marked  divei-genee  from  each  other 
and  from  the  text  published  by  Curtze  that  the  conclusion  seems  legiti- 
mate that  these  were  students'  lecture  notes.  The  shorthand  char- 
acter of  the  writing  further  confirms  this  view,  as  it  shows  that  they 
were  written  largely  for  the  personal  use  of  the  writers. 


SPREAD  OF  THE  NUMERALS  IN  EUROPE   135 

term  Arabic  numerals  became  common.  In  two  places 
there  is  mention  of  the  inventors  of  this  system.  In  the 
introduction  it  is  stated  that  this  science  of  reckoning 
was  due  to  a  philosopher  named  Algus,  whence  the  name 
algorismus,1  and  in  the  section  on  numeration  reference 
is  made  to  the  Arabs  as  the  inventors  of  this  science.2 
While  some  of  the  commentators,  Petrus  de  Dacia3 
among  them,  knew  of  the  Hindu  origin,  most  of  them 
undoubtedly  took  the  text  as  it  stood ;  and  so  the  Arabs 
were  credited  with  the  invention  of  the  system. 

The  first  definite  trace  that  we  have  of  an  algorism 
in  the  French  language  is  found  in  a  manuscript  written 
about  1275.4  This  interesting  leaf,  for  the  part  on  algo- 
rism consists  of  a  single  folio,  was  noticed  by  the  Abbe 
Lebceuf  as  early  as  1741,5  and  by  Daunou  in  1824.6  It 
then  seems  to  have  been  lost  in  the  multitude  of  Paris 
manuscripts;  for  although  Chasles7  relates  his  vain  search 
for  it,  it  was  not  rediscovered  until  1882.  In  that  year 
M.  Ch.  Henry  found  it,  and  to  his  care  we  owe  our  knowl- 
edge of  the  interesting  manuscript.  The  work  is  anony- 
mous and  is  devoted  almost  entirely  to  geometry,  only 

1  "Quidam  philosophus  edidit  nomine  Algus,  unde  et  Algorismus 
nuncupatur."    [Curtze,  loc.  cit.,  p.  1.] 

2  "  Sinistrorsum  autem  scribimus  in  hac  arte  more  arabico  sive 
iudaico,  huius  scientiae  inventorum."  [Curtze,  loc.  cit.,  p.  7.]  The 
Plimpton  manuscript  omits  the  words  "sive  iudaico." 

3  "Non  enim  onmis  numerus  per  quascumque  figuras  Indorum 
repraesentatur,  sed  tantum  determinatus  per  determinatam,  ut  4  non 
per  5,  .  .  ."    [Curtze,  loc.  cit.,  p.  25.] 

4  C.  Henry,  "Sur  les  deux  plus  anciens  trait^s  francais  d'algorisme 
et  de  g&>m<5trie,"  Boncompagni  Bulletino,  Vol.  XV,  p.  49;  Victor 
Mortet,  "Le  plus  ancien  traite"  francais  d'algorisme,"  loc.  cit. 

5  VlZtat  des  sciences  en  France,  depuis  la  mort  du  Roy  Robert,  arrivie 
en  1031,jusqu'a  celle  de  Philippe  le  Bel,  arrivee  en  1314,  Paris,  1741. 

6  Discours  sur  Vital  des  lettres  en  France  au  XIIIe  siecle,  Paris,  1824. 

7  Aperqu  historique,  Paris,  1875  ed.,  p.  464. 


/ 


136  THE  HINDU-ARABIC  NUMERALS 

two  pages  (one  folio)  relating  to  arithmetic.  In  these  the 
forms  of  the  numerals  are  given,  and  a  very  brief  statement 
as  to  the  operations,  it  being  evident  that  the  writer  him- 
self had  only  the  slightest  understanding  of  the  subject. 

Once  the  new  system  was  known  in  France,  even 
thus  superficially,  it  would  be  passed  across  the  Chan- 
nel to  England.  Higden,1  writing  soon  after  the  opening 
of  the  fourteenth  century,  speaks  of  the  French  influence 
at  that  time  and  for  some  generations  preceding :  2  "  For 
two  hundred  years  children  in  scole,  agenst  the  usage 
and  manir  of  all  other  nations  beeth  compelled  for  to 
leave  hire  own  language,  and  for  to  construe  hir  lessons 
and  hire  thynges  hi  Frensche.  .  .  .  Gentilmen  children 
beeth  taught  to  speke  Frensche  from  the  tyme  that  they 
bith  rokked  in  hir  cradell ;  and  uplondissche  men  will 
likne  himself  to  gentylmen,  and  f  ondeth  with  greet  besy- 
nesse  for  to  speke  Frensche." 

The  question  is  often  asked,  why  did  not  these  new 
numerals  attract  more  immediate  attention  ?  Why  did 
they  have  to  wait  until  the  sixteenth  century  to  be  gen- 
erally used  in  business  and  in  the  schools  ?  In  reply  it 
may  be  said  that  in  their  elementary  work  the  schools 
always  wait  upon  the  demands  of  trade.  That  work  which 
pretends  to  touch  the  life  of  the  people  must  come  reason- 
ably near  doing  so.  Now  the  computations  of  business 
until  about  1500  did  not  demand  the  new  figures,  for 
two  reasons:  First,  cheap  paper  was  not  known.  Paper- 
making  of  any  kind  was  not  introduced  into  Europe  until 

1  Ranulf  Higden,  a  native  of  the  west  of  England,  entered  St. 
Werburgh's  monastery  at  Chester  in  1299.  He  was  a  Benedictine 
monk  and  chronicler,  and  died  in  1364.  His  Polychronicon,  a  history 
in  seven  books,  was  printed  by  Caxton  in  1480. 

2  Trevisa's  translation,  Higden  having  written  in  Latin. 


SPREAD  OF  THE  NUMERALS  IN  EUROPE   137 

the  twelfth  century,  and  cheap  paper  is  a  product  of 
the  nineteenth.  Pencils,  too,  of  the  modern  type,  date 
only  from  the  sixteenth  century.  In  the  second  place, 
modern  methods  of  operating,  particularly  of  multiplying 
and  dividing  (operations  of  relatively  greater  importance 
when  all  measures  were  in  compound  numbers  requiring 
reductions  at  every  step),  were  not  yet  invented.  The 
old  plan  required  the  erasing  of  figures  after  they  had 
served  their  purpose,  an  operation  very  simple  with  coun- 
ters, since  they  could  be  removed.  The  new  plan  did 
not  as  easily  permit  this.  Hence  we  find  the  new  numer- 
als very  tardily  admitted  to  the  counting-house,  and  not 
welcomed  with  any  enthusiasm  by  teachers.1 

Aside  from  their  use  in  the  early  treatises  on  the  new 
art  of  reckoning,  the  numerals  appeared  from  time  to 
time  in  the  dating  of  manuscripts  and  upon  monuments. 
The  oldest  definitely  dated  European  document  known 

1  An  illustration  of  this  feeling  is  seen  in  the  writings  of  Prosdocimo 
de' Beldomandi  (b.  c.  1370-1380,  d.  1428):  "Inveni  in  quam  pluribus 
libris  algorismi  nuncupatis  mores  circa  numeros  operandi  satis  varios 
atque  diversos,  qui  licet  boni  existerent  atque  veri  erant,  tamen  f  asti- 
diosi,  turn  propter  ipsarum  regularum  multitudinem,  turn  propter 
earum  deleationes,  turn  etiam  propter  ipsarum  operationum  proba- 
tiones,  utrum  si  bone  fuerint  vel  ne.  Erant  et  etiam  isti  modi  interim 
fastidiosi,  quod  si  in  aliquo  calculo  astroloico  error  contigisset,  calcu- 
latorem  operationem  suam  a  capite  incipere  oportebat,  dato  quod 
error  suus  adhuc  satis  propinquus  existeret;  et  hoc  propter  figuras  in 
sua  operatione  deletas.  Indigebat  etiam  calculator  semper  aliquo 
lapide  vel  sibi  conformi,  super  quo  scribere  atque  faciliter  delere 
posset  figuras  cum  quibus  operabatur  in  calculo  suo.  Et  quia  haec 
omnia  satis  f  astidiosa  atque  laboriosa  mihi  visa  sunt,  disposui  libellum 
edere  in  quo  omnia  ista  abicerentur :  qui  etiam  algorismus  sive  liber 
de  numeris  denominari  poterit.  Scias  tamen  quod  in  hoc  libello  po- 
nere  non  intendo  nisi  ea  quae  ad  calculum  necessaria  sunt,  alia  quae 
in  aliis  libris  practice  arismetrice  tanguntur,  ad  calculum  non  neces- 
saria, propter  brevitatem  dimitendo."  [Quoted  by  A.  Nagl,  Zeitschrift 
fiir  Mathematik  und  Physik,  Hist.-lit.  Abth., Vol.  XXXIV,  p.  143 ;  Smith, 
Bara  Arithmetical  p.  14,  in  facsimile.] 


138  THE  HINDU-ARABIC  NUMERALS 

to  contain  the  numerals  is  a  Latin  manuscript,1  the 
Codex  Vigilanus,  written  in  the  Albelda  Cloister  not 
far  from  Logrofio  in  Spain,  in  976  a.d.  The  nine  char- 
acters (of  gobar  type),  without  the  zero,  are  given  as  an 
addition  to  the  first  chapters  of  the  third  book  of  the 
Origines  by  Isidorus  of  Seville,  in  which  the  Roman  nu- 
merals are  under  discussion.  Another  Spanish  copy  of 
the  same  work,  of  992  a.d.,  contains  the  numerals  in  the 
corresponding  section.  The  writer  ascribes  an  Indian 
origin  to  them  in  the  following  words:  "Item  de  figuris 
arithmetic^.  Scire  debemus  in  Indos  subtil issimum  inge- 
nium  habere  et  ceteras  gentes  eis  in  arithmetica  et  geo- 
metria  et  ceteris  liberalibus  disciplinis  concedere.  Et  hoc 
manifestum  est  in  nobem  figuris,  quibus  designant  unum- 
quemque  gradum  cuiuslibet  gradus.  Quarum  hec  sunt 
forma."  The  nine  jrobar  characters  follow.  Some  of  the 
abacus  forms2  previously  given  are  doubtless  also  of 
the  tenth  century.  The  earliest  Arabic  documents  con- 
taining the  numerals  are  two  manuscripts  of  874  and 
888  A.D.3  They  appear  about  a  century  later  in  a  work 4 
written  at  Shiraz  in  970  a.d.  There  is  also  an  early 
trace  of  their  use  on  a  pillar  recently  discovered  in  a 
church  apparently  destroyed  as  early  as  the  tenth  cen- 
tury, not  far  from  the  Jeremias  Monastery,  in  Egypt. 

1  P.  Ewald,  loc.  cit.  ;  Franz  Steffens,  Lateinische  Palaographie,  pp. 
xxxix-xl.  We  are  indebted  to  Professor  J.  M.  Burnam  fur  a  photo- 
graph of  this  rare  manuscript. 

2  See  the  plate  of  forms  on  p.  88. 

8  Karabacek,  loc.  cit.,  p.  56;  Karpinski,  "Hindu  Numerals  in  the 
Fihrist,"  Bibliotheca  Mathematica,  Vol.  XI  (3),  p.  121. 

4  Woepcke,  " Sur  une  donneV  historique,"  etc..  loc.  cit..  and  "Essai 
d'une  restitution  de  travaux  perdus  d'Apollonius  sur  les  quantity 
irratinnnelles,  d'apres  des  indications  tirees  d'Un  manuscrit  arabe," 
Tome  XIV  des  Mimoires  prisentes  par  divers  savants  a  I'Academie  des 
science*,  Paris,  1850,  uote,  pp.  6-14. 


SPREAD  OF  THE  NUMERALS  IN  EUROPE      139 

A  graffito  in  Arabic  on  this  pillar  has  the  date  349  a.h., 
which  corresponds  to  961  a.d.1  For  the  dating  of  Latin 
documents  the  Arabic  forms  were  used  as  early  as  the 
thirteenth  century.2 

On  the  early  use  of  these  numerals  in  Europe  the 
only  scientific  study  worthy  the  name  is  that  made  by  Mr. 
G.  F.  Hill  of  the  British  Museum.3  From  his  investio-a- 
tions  it  appears  that  the  earliest  occurrence  of  a  date  in 
these  numerals  on  a  coin  is  found  in  the  reign  of  Roger 
of  Sicily  in  1138.4  Until  recently  it  was  thought  that  the 
earliest  such  date  was  1217  a.d.  for  an  Arabic  piece  and 
1388  for  a  Turkish  one.5  Most  of  the  seals  and  medals 
containing  dates  that  were  at  one  time  thought  to  be 
very  early  have  been  shown  by  Mr.  Hill  to  be  of  rela- 
tively late  workmanship.  There  are,  however,  in  Euro- 
pean manuscripts,  numerous  instances  of  the  use  of  these 
numerals  before  the  twelfth  century.  Besides  the  exam- 
ple in  the  Codex  Vigilanus,  another  of  the  tenth  century 
has  been  found  in  the  St.  Gall  MS.  now  in  the  Univer- 
sity Library  at  Zurich,  the  forms  differing  materially  from 
those  in  the  Spanish  codex. 

The  third  specimen  in  point  of  time  in  Air.  Hill's  list  is 
from  a  Vatican  MS.  of  1077.  The  fourth  and  fifth  speci- 
mens are  from  the  Erlangen  MS.  of  Boethius,  of  the  same 

1  Archeological  Report  of  the  Egypt  Exploration  Fund  for  1908-1909, 
London,  1910,  p.  18. 

2  There  was  a  set  of  astronomical  tables  in  Boncompagni's  library 
bearing  this  date:  "Nota  quod  anno  dni  firi  ihii  xpi.  1264.  perfecto." 
See  Narducci's  Catalogo,  p.  130. 

3  "On  the  Early  use  of  Arabic  Numerals  in  Europe,"  read  before 
the  Society  of  Antiquaries  April  14, 1910,  and  published  in  Archccologia 
in  the  same  year. 

4  Ibid.,  p.  8,  n.    The  date  is  part  of  an  Arabic  inscription. 

5  O.  Codrington,  A  Manual  of  Musalman  Numismatics,  London, 
1904. 


140 


THE  HINDU-ARABIC  NUMERALS 


(eleventh)  century,  and  the  sixth  and  seventh  are  also 
from  an  eleventh-century  MS.  of  Boethius  at  Chartres. 

Earliest  Manuscript  Forms 


1 

I 

■  ■■■■'   ' 

X 

t 

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M 

lr. 

1 

8 

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2 

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V 

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W-i 

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Ja 

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A 

8 

2 

1077 

4 

I 

TT 

JZ 

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Q 

b 

^v 

8 

9 

a. 

5 

I 

V 

? 

oC 

4 

k) 

A 

8 

9 

© 

s 

6 

I 

z 

fr 

B 

M 

13 

\ 

ff 

<r 

© 

3 

7 

I 

F 

X 

e£ 

•q 

F 

K 

8 

9 

ft 

/ 

r 

* 

£-* 

9 

Z7 

A 

2 

/ 

XtwXIt 

9 

i 

TT 

<N 

* 

* 

Tg 

V 

8 

& 

l^SE 

10 

t 

5 

%* 

q 

p 

t 

CO 

II 

jt 

* 

Xfc 

<b 

j? 

fc 

V 

6 

Tb 

^L? 

12. 

/ 

2? 

$ 

<# 

H 

b 

V 

r 

3 

SE 

lb 

1 

5" 

1*1 

£ 

h 

la 

V 

8 

b 

x  i») 

M»ai 

14- 

I 

S 

H? 

j8 

I? 

b 

V 

B 

& 

c.1200 

IS 

I 

Z 

^ 

* 

4 

fa 

A 

& 

^ 

C.IJOO 

16 

1 

z 

u 

°e 

4 

L 

X 

8 

9 

? 

•7 

z 

X 

vL. 

a 

lo 

^ 

8 

1 

J 

18 

I 

z 

^<* 

<*» 

q 

bin 

A 

8 

P 

t 

•9 

/ 

V 

? 

^ 

^ 

/> 

A 

8 

} 

0 

>V1 

. 

20 

J 

s 

A 

s-« 

v 

& 

A 

Z 

3 

*£ 

2t 

I 

V 

^ 

RF 

i 

Lr 

A 

3 

9 

0 

xvTo 

^ 

These  and  other  early  forms  are  given  by  Mr.  Hill  in  this 
table,  which  is  reproduced  with  his  kind  permission. 

This  is  one  of  more  than   fifty  tables  given  in  Mr. 
Hill's  valuable  paper,  and  to  this  monograph  students 


SPREAD  OF  THE  NUMERALS  IN  EUROPE   141 

are  referred  for  details  as  to  the  development  of  number- 
forms  in  Europe  from  the  tenth  to  the  sixteenth  cen- 
tury. It  is  of  interest  to  add  that  he  has  found  that 
among  the  earliest  dates  of  European  coins  or  medals 
in  these  numerals,  after  the  Sicilian  one  already  men- 
tioned, are  the  following :  Austria,  1484  ;  Germany,  1489 
(Cologne)  ;  Switzerland,  1424  (St.  Gall)  ;  Netherlands, 
1474;  France,  1485;  Italy,  1390.1 

The  earliest  English  coin  dated  in  these  numerals  was 
struck  in  1551,2  although  there  is  a  Scotch  piece  of  1539. 3 
In  numbering  pages  of  a  printed  book  these  numerals 
were  first  used  in  a  work  of  Petrarch's  published  at  Co- 
logne in  1471.4  The  date  is  given  in  the  following  form 
in  the  Biblia  Pauperumf  a  block-book  of  1470,  while  in 

another  block-book  which  possibly  goes  back  to  c.  1430  6 
the  numerals  appear  in  several  illustrations,  with  forms 
as  follows : 

Y 

Many  printed  works  anterior  to  1471  have  pages  or  chap- 
ters numbered  by  hand,  but  many  of  these  numerals  are 

1  See  Arbuthnot,  The  Mysteries  of  Chronology,  London,  1000,  pp.  75, 
78,  08  ;  F.  Pichler,  Eepertorium  der  steierischen  Miinzkunde,  Gratz,  1875, 
where  the  claim  is  made  of  an  Austrian  coin  of  1458 ;  Bibliothecu 
Mathematica,  Vol.  X  (2),  p.  120,  and  Vol.  XII  (2),  p.  120.  There  is  a 
Brabant  piece  of  1478  in  the  collection  of  D.  E.  Smith. 

2  A  specimen  is  in  the  British  Museum.    [Arbuthnot,  p.  79.] 

3  Ibid.,  p.  79. 

4  Liber  de  Bemediis  idriusque  fortunae  Coloniae. 

5  Fr.  Walthern  et  Hans  Hurning,  Nordlingen. 

6  Ars  Memorandi,  one  of  the  oldest  European  block-books. 


142  THE  HIXDU-ARABIC  NUMERALS 

of  date  much  later  than  the  printing  of  the  work.  Other 
works  were  probably  numbered  directly  after  printing. 
Thus  the  chapters  2,  3,  4,  5,  6  in  a  book  of  1470  l  are 
numbered  as  follows :  Capitulem  zm.,  .  .  .  5m.,  .  .  .  4111., 
...  v,  ...  vi,  and  followed  by  Roman  numerals. 
This  appears  in  the  body  of  the  text,  in  spaces  left  by 
the  printer  to  be  filled  in  by  hand.  Another  book2  of 
1470  has  pages  numbered  by  hand  with  a  mixture  of 
Roman  and  Hindu  numerals,  thus, 

Q  2_  ""7  for  125  £  U  O  for  150 

£   Q  /^  for  147  ^Q  7*.   for  202 

As  to  monumental  inscriptions,3  there  was  once 
thought  to  be  a  gravestone  at  Katharein,  near  Troppau, 
with  the  date  1007,  and  one  at  Biebrich  of  1299.  There 
is  no  doubt,  however,  of  one  at  Pforzheim  of  1371 
and  one  at  Ulm  of  1388.4  Certain  numerals  on  Wells 
Cathedral  have  been  assigned  to  the  thirteenth  century, 
but  they  are  undoubtedly  considerably  later.5 

The  table  on  page  143  will  serve  to  supplement  that 
from  Mr.  Hill's  work.6 

1  Eusebius  Caesariensis,  Be  praeparatione  evangelica,  Venice,  Jenson, 
1470.  The  above  statement  holds  for  copies  in  the  Astor  Library  and 
in  the  Harvard  University  Library. 

2  Francisco  de  Retza,  Comestorium  vitiorum,  Nurnberg,  1470.  The 
copy  referred  to  is  in  the  Astor  Library. 

3  See  Mauch,  "  Leber  den  Gebrauch  arabischer  Ziffern  und  die 
Veranderungen  derselben,"  Anzeiger  fur  Kunde  der  deutschen  Vorzeit, 
1801,  columns  40,  81,  110,  151,  189,  229,  and  208;  Calmet,  Recherches 
sur  Vorigine  des  chiffres  d'arithme'tique,  plate,  loc.  eit. 

4  Giinther,  Geschichte,  p.  175,  n.;  Mauch,  loc.  cit. 

5  These  are  given  by  W.  R.  Lethaby,  from  drawings  by  J.  T.  Irvine, 
in  the  Proceedings  of  the  Society  of  Antiquaries,  1900,  p.  200. 

6  There  are  some  ill-tabulated  forms  to  be  found  in  J.  Bowring, 
The  Decimal  St/stern,  London,  1854,  pp.  23,  25,  and  in  L.  A.  Chassant, 
Dictionnaire  des  abreviations  latines  et  francaises  .  .  .  du  moyen  age, 


SPREAD  OF  THE  NUMERALS  IX  EUROPE      143 


Early  Manuscript  Forms 
12  3        4       567890 

0  *   7   ^   A   <f  <r  A    %  f>  p 
1~2     >   A  <t    <TA    S  9  0 


Twelfth  century 
1197  A. n. 

1275  A. d. 
c.  1204  A.D. 
c.  1303  A.D. 

c.  1360  A.D. 
e.  1442  A.D. 


Paris,  mdccclxvi,  p.  113.  The  best  sources  we  have  at  present, 
aside  from  the  Hill  monograph,  are  P.  Treutlein,  Geschichte  unserer 
Zahlzeichen,  Karlsruhe,  1875;  Cantor's  Geschichte,  Vol.  I,  table;  M. 
Pruu,  Manuel  de  paleographie  latine  et  franqaise,  2d  ed.,  Paris,  1892, 
p.  164;  A.  Cappelli,  Bizionario  di  abbreviature  latine  ed  italiane, 
Milan,  1899.  An  interesting  early  source  is  found  in  the  rare  Caxton 
work  of  1480,  The  Myrrour  of  the  World.  In  Chap.  X  is  a  cut  with 
the  various  numerals,  the  chapter  beginning  "The  fourth  scyence  is 
called  arsmetrique."  Two  of  the  fifteen  extant  copies  of  this  work 
are  at  present  in  the  library  of  Mr.  J.  P.  Morgan,  in  New  York. 

a  From  the  twelfth-century  manuscript  on  arithmetic,  Curtze,  loc. 
cit.,  Abhandlungen,  and  Xagl,  loc.  cit.  The  forms  are  copied  from 
Plate  VII  in  Zeitschrift  fur  Matkematik  und  Physik,  Vol.  XXXIV. 

b  From  theRegensburg  chronicle.  Plate  containing  some  of  these 
numerals  in  Monumenta  Germaniae  historica,  "Scriptores"Vol.  XVII, 
plate  to  p.  184;  Wattenbach,  Anleitung  zur  lateinischen  Palaeographie, 
Leipzig,  1886,  p.  102  ;  Boehmer,  Fontes  rerum  Germanicarum,  Vol.  Ill, 
Stuttgart,  1852,  p.  lxv. 

c  French  Algorismus  of  1275  ;  from  an  unpublished  photograph  of 
the  original,  in  the  possession  of  D.  E.  Smith.    See  also  p.  135. 

d  From  a  manuscript  of  Boethius  c.  1294,  in  Mr.  Plimpton's  library. 
Smith,  Kara  Arithmetica,  Plate  I. 

e  Xumerals  in  a  1303  manuscript  in  Sigmaringen,  copied  from 
Wattenbach,  loc.  cit.,  p.  102. 

f  From  a  manuscript,  Add.  Manuscript  27,589,  British  Museum, 
1360  a.d.  The  work  is  a  computus  in  which  the  date  1360  appears, 
assigned  in  the  British  Museum  catalogue  to  the  thirteenth  century. 

g  From  the  copy  of  Sacrobosco's  Algorismus  in  Mr.  Plimpton's 
library.    Date  c.  1442.    See  Smith,  liara  Aritlanetica,  p.  450. 


144 


THE  HINDU-ARABIC  NUMERALS 


For  the  sake  of  further  com- 
parison, three  illustrations  from 
works  in  Mr.  Plimpton's  library, 
reproduced  from  the  Rara  Arith- 
metical may  be  considered.  The 
J  n;  first  is  from  a  Latin  manuscript 

Y  *  '  on  arithmetic,1  of  which  the  orig- 

inal was  written  at  Paris  in  1424 
by  Rollandus,  a  Portuguese  phy- 
sician, who  prepared  the  work  at 
the  command  of  John  of  Lan- 
caster, Duke  of  Bedford,  at  one 
time  Protector  of  England  and 
Regent  of  France,  to  whom  the 
work  is  dedicated.  The  figures 
show  the  successive  powers  of  2. 
The  second  illustration  is  from 
Luca  da  Firenze's  Inprencipio 
darte  dabacho,2  c.  1475,  and  the 
third  is  from  an  anonymous  manu- 
script3 of  about  1500. 

As  to  the  forms  of  the  num- 
erals, fashion  played  a  leading 
part  until  printing  was  invented.  This  tended  to  fix  these 
forms,  although  in  writing  there  is  still  a  great  variation, 


Vr*rtVi<r?|. 


»!• 


<1_    ^  .  f  •  c 


8 


as  witness   the    French   5  and  the   German    7  and   9. 
Even    in    printing    there    is    not    complete    uniformity, 


1 

.i. 

.?■ 

■4" 

■(,■ 

.  err 

n~ a- 

0- 

0 

(O 

II 

4-   ... 

8- 

.   12- 

.  \C~ 

T-O 

7-<f' 

^0- 

■?*-■ 

■<}«=* 

40. 

Oft. 

3 

■err 

0- 

.|Z~ 

.If,. 

18- 

.«-!■ 

.%Ar 

■  2-A 

w 

?? 

1  See  Rara  Arithmetical  pp.  446-447. 

2  Ibid.,  pp.  40D-470.  3  Ibid.,  pp.  477-478. 


SPREAD  OF  THE  NUMERALS  IN  EUROPE      145 

and  it  is  often  difficult  for  a  foreigner  to  distinguish 
between  the  3  and  5  of  the  French  types. 

As  to  the  particular  numerals,  the  following  are  some 
of  the  forms  to  be  found  in  the  later  manuscripts  and 
in  the  early  printed  books. 

1.  In  the  early  printed  books  "one"  was  often  i,  perhaps 
to  save  types,  just  as  some  modern  typewriters  use  the 
same  character  for  1  and  l.1  In  the  manuscripts  the  "  one  " 
appears  in  such  forms  as  2 

2.  "Two"  often  appears  as  z  in  the  early  printed  books, 
12  appearing  as  iz.3  In  the  medieval  manuscripts  the 
following  forms  are  common  :  4 

1  The  i  is  used  for  "  one  "  in  the  Treviso  arithmetic  (1478),  Clichto- 
veus  (c.  1507  ed.,  where  both  i  and  j  are  so  used),  Chiarini  (1481), 
Sacrobosco  (1488  ed.),  and  Tzwivel  (1507  ed.,  where  jj  and  jz  are  used 
for  11  and  12).  This  was  not  universal,  however,  for  the  Algorithmus 
Unealis  of  c.  1488  has  a  special  type  for  1.  In  a  student's  notebook  of 
lectures  taken  at  the  University  of  Wiirzburg  in  1660,  in  Mr.  Plimpton's 
library,  the  ones  are  all  in  the  form  of  i. 

2  Thus  the  date  J.J  ~Cj  &',  for  1580,  appears  in  a  MS.  in  the  Lau- 
rentian  library  at  Florence.  The  second  and  the  following  five  char- 
acters are  taken  from  Cappelli's  Dizionario,  p.  380,  and  are  from 
manuscripts  of  the  twelfth,  thirteenth,  fourteenth,  sixteenth,  seven- 
teenth, and  eighteenth  centuries,  respectively. 

3  E.  g.  Chiarini's  work  of  1481 ;  Clichtoveus  (c.  1507). 

4  The  first  is  from  an  algorismus  of  the  thirteenth  century,  in  the 
Hannover  Library.  [See  Gerhardt,  "  Ueber  die  Entstehung  und 
Ausbreitung  des  dekadischen  Zahlensystems,"  loc.  cit.,  p.  28.]  The 
second  character  is  from  a  French  algorismus,  c.  1275.  [Boncom- 
pagni  Bulletin/),  Vol.  XV,  p.  51.]  The  third  and  the  following  sixteen 
characters  are  given  by  Cappelli,  loc.  cit.,  and  are  from  manuscripts 
of  the  twelfth  "(1),  thirteenth  (2),  fourteenth  (7),  fifteenth  (3),  six- 
teenth (1),  seventeenth  (2),  and  eighteenth  (1)  centuries,  respectively. 


146  THE  HINDU-ARABIC  NUMERALS 

It  is  evident,  from  the  early  traces,  that  it  is  merely 
a  cursive  form  for  the  primitive  =,  just  as  3  comes  from 
=  ,  as  in  the  Nana  Ghat  inscriptions. 

3.  "  Three  "  usually  had  a  special  type  in  the  first  printed 
books,  although  occasionally  it  appears  as  ^.l  In  the 
medieval  manuscripts  it  varied  rather  less  than  most  of 
the  others.    The  following  are  common  forms :  2 

4.  "  Four  "  has  changed  greatly  ;  and  one  of  the  first 
tests  as  to  the  age  of  a  manuscript  on  arithmetic,  and 
the  place  where  it  was  written,  is  the  examination 
of  this  numeral.  Until  the  time  of  printing  the  most 
common  form  was  X,  although  the  Florentine  manu- 
script of  Leonard  of  Pisa's  work  has  the  form  /^.  ;3 
but  the  manuscripts  show  that  the  Florentine  arithme- 
ticians and  astronomers  rather  early  began  to  straighten 
the  first  of  these  forms  up  to  forms  like  9"  4  and  $-  4 
or  <)-  ,5  more  closely  resembling  our  own.  The  first 
printed  books  generally  used  our  present  form 6  with  the 
closed  top  4  >  tne  °Pen  t0P  used  in  writing  00  bemg 

i  Thus  Chiarini  (1481)  has  Z3  for  23. 

2  The  first  of  these  is  from  a  French  algorismus,  c.  1275.  The 
second  and  the  following  eight  characters  are  given  by  Cappelli, 
loc.  cit.,  and  are  from  manuscripts  of  the  twelfth  (2),  thirteenth, 
fourteenth,  fifteenth  (3),  seventeenth,  and  eighteenth  centuries, 
respectively. 

8  See  Nagl,  loc.  cit. 

4  Hannover  algorismus.  thirteenth  century. 

5  See  the  Dagomari  manuscript,  in  liara  Arithmetica,  pp.  435, 
437-440. 

6  But  in  the  woodcuts  of  the  Margarita  Philowphka  (1503)  the  old 
forms  are  used,  although  the  new  ones  appear  in  the  text.  In  Caxton's 
Myrruur  of  the  World  (1480)  the  old  form  is  used. 


SPREAD  OF  THE  NUMERALS  IN  EUROPE      147 

purely  modern.    The  following  are  other  forms  of  the 
four,  from  various  manuscripts  : x 

5.  "  Five  "  also  varied  greatly  before  the  time  of  print- 
ing.   The  following  are  some  of  the  forms : 2 

6.  "  Six  "  has  changed  rather  less  than  most  of  the 
others.  The  chief  variation  has  been  in  the  slope  of  the 
top,  as  will  be  seen  in  the  following : 3 

^IQ.c,  &,<?,&-;  %• 

7.  "  Seven,"  like  "  four,"  has  assumed  its  present  erect 
form  only  since  the  fifteenth  century.  In  medieval 
times  it  appeared  as  follows : 4 

A,  a, si,  t,/r.A,nA,*.\)a 

1  Cappelli,  loc.  cit.  They  are  partly  from  manuscripts  of  the  tenth, 
twelfth,  thirteenth  (3),  fourteenth  (7),  fifteenth  (6),  and  eighteenth 
centuries,  respectively.  Those  in  the  third  line  are  from  Chassant's 
Dictionnaire,  p.  113,  without  mention  of  dates. 

2  The  first  is  from  the  Hannover  algorismus,  thirteenth  century. 
The  second  is  taken  from  the  Rollandus  manuscript,  1424.  The 
others  in  the  first  two  lines  are  from  Cappelli,  twelfth  (3),  fourteenth 
(5),  fifteenth  (13)  centuries,  respectively.  The  third  line  is  from 
Chassant,  loc.  cit.,  p.  113,  no  mention  of  dates. 

3  The  first  of  these  forms  is  from  the  Hannover  algorismus,  thir- 
teenth century.  The  following  are  from  Cappelli,  fourteenth  (3),  fif- 
teenth, sixteenth  (2),  and  eighteenth  centuries,  respectively. 

4  The  first  of  these  is  taken  from  the. Hannover  algorismus,  thir- 
teenth  century.    The  following  forms  are  from  Cappelli,   twelfth, 


148  THE  HINDU-ARABIC  NUMERALS 

8.  "Eight,"  like  "six,"  has  changed  but  little.  In 
medieval  times  there  are  a  few  variants  of  interest  as 
follows : * 

ft,  •&;,&, 5,  tf 

In  the  sixteenth  century,  however,  there  was  mani- 
fested a  tendency  to  write  it  Co2 

9.  "  Nine "  has  not  varied  as  much  as  most  of  the 
others.    Among  the  medieval  forms  are  the  following : 3 

0.  The  shape  of  the  zero  also  had  a  varied  history. 
The  following  are  common  medieval  forms  : 4 

The  explanation  of  the  place  value  was  a  serious  mat- 
ter to  most  of  the  early  writers.  If  they  had  been  using 
an  abacus  constructed  like  the  Russian  chotii,  and  had 
placed  this  before  all  learners  of  the  positional  system, 
there  would  have  been  little  trouble.    But  the  medieval 

thirteenth,  fourteenth  (5),  fifteenth(2),  seventeenth,  and  eighteenth 
centuries,  respectively. 

1  All  of  these  are  given  by  Cappelli,  thirteenth,  fourteenth,  fifteenth 
(2),  and  sixteenth  centuries,  respectively. 

2  Smith,  liara  Arithmetical  p.  489.  This  is  also  seen  in  several  of  the 
Plimpton  manuscripts,  as  in  one  written  at  Ancona  in  1684.  See  also 
Cappelli,  loc.  cit. 

8  French  algorismus,  c.  1275,  for  the  first  of  these  forms.  Cap- 
pelli, thirteenth,  fourteenth,  fifteenth  (3),  and  seventeenth  centuries, 
respectively.  The  last  three  are  taken  from  Byzantinische  Analekten, 
J.  L.  Heiberg,  being  forms  of  the  fifteenth  century,  but  not  at  all 
common.    9  was  the  old  Greek  symbol  for  90. 

4  For  the  first  of  these  the  reader  is  referred  to  the  forms  ascribed 
to  Boethius,  in  the  illustration  on  p.  88  ;  for  the  second,  to  Radulph 
of  Laon,  see  p.  60.  The  third  is  used  occasionally  in  the  Rollandus 
(1424)  manuscript,  in  Mr.  Plimpton's  library.  The  remaining  three 
are  from  Cappelli,  fourteenth  (2)  and  seventeenth  centuries. 


SPREAD  OF  THE  NUMERALS  IN  EUROPE   149 

line-reckoning,  where  the  lines  stood  for  powers  of  10 
and  the  spaces  for  half  of  such  powers,  did  not  lend 
itself  to  this  comparison.  Accordingly  we  find  such 
labored  explanations  as  the  following,  from  TJie  Crafte 
of  Nornbrynge : 

"  Euery  of  these  figuris  bitokens  hym  self  e  &  no  more, 
yf  he  stonde  in  the  first  place  of  the  rewele.  .  .  . 

"  If  it  stonde  in  the  secunde  place  of  the  rewle,  he  be- 
tokens ten  tymes  hym  selfe,  as  this  figure  2  here  20 
tokens  ten  tyme  hym  selfe,  that  is  twenty,  for  he  hym 
selfe  betokens  tweyne,  &  ten  tymes  twene  is  twenty. 
And  for  he  stondis  on  the  lyft  side  &  in  the  secunde 
place,  he  betokens  ten  tyme  hym  selfe.  And  so  go 
forth.  .  .  . 

"  Nil  cifra  significat  sed  dat  signare  sequenti.  Expone 
this  verse.  A  cifre  tokens  no3t,  bot  he  makes  the  figure 
to  betoken  that  comes  after  hym  more  than  he  shuld  & 
he  were  away,  as  thus  10.  here  the  figure  of  one  tokens 
ten,  &  yf  the  cifre  were  away  &  no  figure  byf ore  hym  he 
schuld  token  bot  one,  for  than  he  schuld  stonde  in  the 
first  place.  .  .  ."  1 

It  would  seem  that  a  system  that  was  thus  used  for 
dating  documents,  coins,  and  monuments,  would  have 
been  generally  adopted  much  earlier  than  it  was,  par- 
ticularly in  those  countries  north  of  Italy  where  it  did 
not  come  into  general  use  until  the  sixteenth  century. 
This,  however,  has  been  the  fate  of  many  inventions,  as 
witness  our  neglect  of  logarithms  and  of  contracted  proc- 
esses to-day. 

As  to  Germany,  the  fifteenth  century  saw  the  rise  of 
the  new  symbolism  ;  the  sixteenth  century  saw  it  slowly 
1  Smith,  An  Early  English  Algorism. 


150  THE  HINDU-ARABIC   NUMERALS 

gain  the  mastery ;  the  seventeenth  century  saw  it  finally 
conquer  the  system  that  for  two  thousand  years  had 
dominated  the  arithmetic  of  business.  Not  a  little  of 
the  success  of  the  new  plan  was  due  to  Luther's  demand 
that  all  learning  should  go  into  the  vernacular.1 

During  the  transition  period  from  the  Roman  to  the 
Arabic  numerals,  various  anomalous  forms  found  place. 
For  example,  we  have  in  the  fourteenth  century  ca  for 
104;2  1000.  300.  80  et  4  for  1384;3  and  in  a  manuscript 
of  the  fifteenth  century  12901  for  1291.4  In  the  same 
century  m.cccc.8II  appears  for  1482,5  while  M°CCCC°50 
(1450)  and  MCCCCXL6  (1446)  are  used  by  Theodo- 
ricus  Ruffi  about  the  same  time.6  To  the  next  century 
belongs  the  form  lvojj  for  1502.  Even  in-  Sfortunati's 
Nuovo  lume7  the  use  of  ordinals  is  quite  confused,  the 
propositions  on  a  single  page  being  numbered  "  tertia," 
"  4,"  and  "  V." 

Although  not  connected  with  the  Arabic  numerals  in 
any  direct  way,  the  medieval  astrological  numerals  may 
here  be  mentioned.  These  are  given  by  several  early 
writers,  but  notably  by  Noviomagus  (1539),8  as  follows  9: 

12  3  4  5  6  7  8  9  10 

1 '  —D  — q  ^T3  -?*  — c-  — ^  — y  — r  ■ — 1 

1  Kuckuck,  p.  5.  2  A.  Cappelli,  loc.  cit.,  p.  372. 

3  Smith,  Rara  Arithmetica,  p.  443. 

4  Curtze,  Petri  PhUomeni  de  Dacia  etc.,  p.  ix. 

5  Cappelli,  loc.cit.,  p.  376.  6  Curtze,  loc.  cit.,  pp.  vm-ix,  note. 

7  Edition  of  1544-1545,  f .  52. 

8  De  numeris  libri  II,  1544  ed.,  cap.  xv.  Heilbronner,  loc.  cit.,  p. 
736,  also  gives  them,  and  compares  this  with  other  systems. 

9  Noviomagus  says  of  them  :  "  De  quibusdam  Astrologicis,  sive 
Chaldaicis  numerorum  notis.  .  .  .  Sunt  &  alias  qusedam  notas,  quibus 
Chaldaei  &  Astrologii  quemlibet  numerum  artificiose  &  argute  descri- 
bunt,  scitu  periucundae,  quas  nobis  communicauit  Rodolphus  Paluda- 
nus  Nouiomagus." 


SPREAD  OF  THE  NUMERALS  IN  EUROPE      151 

Thus  we  find  the  numerals  gradually  replacing  the 
Roman  forms  all  over  Europe,  from  the  time  of  Leo- 
nardo of  Pisa  until  the  seventeenth  century.  But  in  the 
Far  East  to-day  they  are  quite  unknown  in  many  coun- 
tries, and  they  still  have  their  way  to  make.  In  many 
parts  of  India,  among  the  common  people  of  Japan  and 
China,  in  Siam  and  generally  about  the  Malay  Peninsula, 
in  Tibet,  and  among  the  East  India  islands,  the  natives 
still  adhere  to  their  own  numeral  forms.  Only  as  West- 
ern civilization  is  making  its  way  into  the  commercial 
life  of  the  East  do  the  numerals  as  used  by  us  find  place, 
save  as  the  Sanskrit  forms  appear  in  parts  of  India.  It 
is  therefore  with  surprise  that  the  student  of  mathematics 
comes  to  realize  how  modern  are  these  forms  so  common 
in  the  West,  how  limited  is  their  use  even  at  the  present 
time,  and  how  slow  the  world  has  been  and  is  in  adopt- 
ing such  a  simple  device  as  the  Hindu-Arabic  numerals. 


INDEX 


Abbo  of  Fleury,  122 
'Abdallah  ibn  al-Hasan,  92 
'Abdallatif  ibn  Yttsuf,  93 
'Abdalqadir  ibn 'All  al-Sakhawi,  6 
Abenragel,  34 
Abraham  ibn  Mei'r  ibn  Ezra,  see 

Rabbi  ben  Ezra 
Abu  'All  al-Hosein  ibn  Sina,  74 
Abu  '1-Hasan,  93,  100 
Abu  '1-Qasim,  92 
Abu  '1-Teiyib,  97 
Abii  Nasr,  92 
Abu  Roshd,  113 
Abu  Sahl  Dunash  ibn  Tamiin,  65, 

67 
Adelhard  of  Bath,  5,  55,  97,  119, 

123,  126 
Adhemar  of  Chabanois,  111 
Ahmed  al-NasawI,  98 
Ahmed  ibn  'Abdallah,  9,  92 
Ahmed  ibn  Mohammed,  94 
Ahmed  ibn  'Omar,  93 
Aksaras,  32 
Alanns  ab  Insulis,  124 
Al-BagdadI,  93 
Al-BattanI,  54 
Albelda  (Albaida)  MS.,  116 
Albert,  J.,  62 
Albert  of  York,  103 
Al-Blruni,  6,  41,  49,  65,  92,  93 
Alcuin,  103 

Alexander  the  Great,  76 
Alexander  de  Villa  Dei,  11,  133 
Alexandria,  64,  82 


Al-Fazari,  92 

Alfred,  103 

Algebra,  etymology,  5 

Algerian  numerals,  68 

Algorism,  97 

Algorismus,  124,  126,  135 

Algorismus  cifra,  120 

Al-Hassar,  65 

'All  ibn  Abi  Bekr,  6 

'All  ibn  Ahmed,  93,  98 

Al-Karabisi,  93 

Al-Khowarazmi,  4,  9,  10,  92,  97, 

98,  125,  126 
Al-Kindl,  10,  92 
Almagest,  54 
Al-Magrebi,  93 
Al-Mahalli,  6 
Al-Mamun,  10,  97 
Al-Mansur,  96,  97 
Al-Mas'udI,  7,  92 
Al-Nadim,  9 
Al-NasawI,  93,  98 
Alphabetic  numerals,  39,  40,  43 
Al-Qasim,  92 
Al-Qass,  94 
Al-Sakhawl,  6 
Al-SardafI,  93 
Al-Sijzi,  94 
Al-SufI,  10,  92 
Ambrosoli,  118 
Ahkapalli,  43 
Apices,  87,  117,  118 
Arabs,  91-98 
Arbuthnot,  141 


153 


154 


THE  HINDU-ARABIC  NUMERALS 


Archimedes,  15,  16 
Arcus  Pictagore,  122 
Arjuna,  15 
Arnold,  E.,  15,  102 
Ars  memoranda,  141 
Aryabhata,  39,  43,  44 
Aryan  numerals,  19 
Aschbach,  134 
Ashmole,  134 
Asoka,  19,  20,  22,  81 
As-sifr,  57,  58 
Astrological  numerals,  150 
Atharva-Veda,  48,  50,  55 
Augustus,  80 
Averroes,  113 
Avicenna,  58,  74,  113 

Babylonian  numerals,  28 

Babylonian  zero,  51 

Bacon,  R.,  131 

Bactrian  numerals,  19,  30 

Bseda,  2,  72 

Bagdad,  4,  96 

Bakhsali  manuscript,  43, 49,  52,  53 

Ball,  C.  J.,  35 

Ball,  W.  W.  R.,  36,  131 

Bana,  44 

Barth,  A.,  39 

Bayang  inscriptions,  39 

Bayer,  33 

Bayley,  E.  C,  19,  23, 30, 32,  52,  89 

Beazley,  75 

Bede,  see  Baeda 

Beldomandi,  137 

Beloch,  J.,  77 

Bendall,  25,  52 

Benfey,  T.,  26 

Bernelinus,  88,  112,  117,  121 

Besagne,  128 

Besant,  W.,  109 

Bettino,  36 

Bhandarkar,  18,  47,  50 


Bhaskara,  53,  55 

Biernatzki,  32 

Biot,  32 

Bjornbo,  A.  A.,  125,  126 

Blassiere,  119 

Bloomfleld,  48 

Blume,  85 

Boeckh,  62 

Boehmer,  143 

Boeschenstein,  119 

Boethius,  63,  70-73,  83-90 

Boissiere,  63 

Bombelli,  81 

Bonaini,  128 

Boncompagni,  5,  6, 10,  48,  50, 123, 

125 
Borghi,  59 
Borgo,  119 
Bougie,  130 
Bowring,  J.,  56 
Brahmagupta,  52 
Brahmanas,  12,  13 
Brahml,  19,  20,  31,  83 
Brandis,  J.,  54 
Brhat-Samhita,  39,  44,  78 
Brockhaus,  43 
Bubnov,  65,  84,  110,  116 
Buddha,  education  of,  15,  16 
Btidinger,  110 
Bugia,  130 

Buhler,  G.,  15,  19,  22,  31,  44,  50 
Burgess,  25 
Biirk,  13 

Burmese  numerals,  36 
Burnell,  A.  C,  18,  40 
Buteo,  61 

Calandri,  59,  81 

Caldwell,  R.,  19 

Calendars,  133 

Calmet,  34 

Cantor,  M.,  5,  13,  30,  43,  84 


INDEX 


155 


Capella,  86 

Cappelli,  143 

Caracteres,  87,  113,  117,  119 

Cardan,  119 

Carmen  de  Algorismo,  11,  134 

Casagrandi,  132 

Casiri,  8,  10 

Cassiodorus,  72 

Cataldi,  62 

Cataneo,  3 

Caxton,  143,  146 

Ceretti,  32 

Ceylon  numerals,  36 

Chalfont,  F.  H.,  28 

Champenois,  60 

Characters,  see  Caracteres 

Charlemagne,  103 

Chasles,    54,    60,    85,    116,    122, 

135 
Chassant,  L.  A.,  142 
Chaucer,  121 
Chiarini,  145,  146 
Chiffre,  58 

Chinese  numerals,  28,  56 
Chinese  zero,  56 
Cifra,  120,  124 
Cipher,  58 
Circulus,  58,  60 
Clichtoveus,  61,  119,  145 
Codex  Vigilanus,  138 
Codrington,  O.,  139 
Coins  dated,  141 
Colebrooke,  8,  26,  46,  53 
Constantine,  104,  105 
Cosmas,  82 
Cossali,  5 
Counters,  117 
Courteille,  8 
Coxe,  59 

Crafte  of  Nombrynge,  11,  87, 149 
Crusades,  109 
Cunningham,  A.,  30,  75 


Curtze,  55,  59,  126,  134 
Cyfra,  55 

Dagomari,  146 
D'Alviella,  15 
Dante,  72 

Dasypodius,  33,  57,  63 
Daunou,  135 
Delambre,  54 
Devanagari,  7 
Devoulx,  A.,  68 
Dhruva,  49,  50 
Dicasarchus  of  Messana,  77 
Digits,  119 
Diodorus  Siculus,  76 
Du  Cange,  62 
Dumesnil,  36 

Dutt,  R.  C,  12,  15,  18,  75 
Dvivedl,  44 

East  and  "West,  relations,  73-81, 

100-109 
Egyptian  numerals,  27 
Eisenlohr,  28 
Elia  Misrachi,  57 
Enchiridion  Algorismi,  58 
Enestrom,  5,  48,  59,  97,  125,  128 
Europe,  numerals  in,  63,  99,  128, 

136 
Eusebius  Caesariensis,  142 
Euting,  21 
Ewald,  P.,  116 

Fazzari,  53,  54 

Fibonacci,  see  Leonardo  of  Pisa 

Figura  nihili,  58 

Figures,  119.    See  numerals. 

Fihrist,  67,  68,  93 

Finaeus,  57 

FirdusI,  81 

Fitz  Stephen,  W.,  109 

Fleet,  J.  C,  19,  20,  50 


156 


THE  HINDU-ARABIC  NUMERALS 


Floras,  80 

Fliigel,  G.,  68 

Francisco  cle  Retza,  142 

Francois,  58 

Friedlein,  G.,  84,  113,  116,  122 

Froude,  J.  A.,  129 

Gandhara,  19 

Garbe,  48 

Gasbarri,  58 

Gantier  de  Coincy,  120,  124 

Gemma  Frisius,  2,  3,  119 

Gerber,  113 

Gerbert,  108,  110-120,  122 

Gerhardt,  C.  I.,  43,  56,  93,  118 

Gerland,  88,  123 

Gherard  of  Cremona,  125 

Gibbon,  72 

Giles,  H.  A.,  79 

Ginanni,  81 

Giovanni  di  Danti,  58 

Glareanns,  4,  119 

Gnecchi,  71,  117 

Gobar   numerals,    65,    100,    112, 

124,  138 
Gow,  J.,  81 
Grammateus,  61 
Greek  origin,  33 
Green,  J.  R.,  109 
Greenwood,  I.,  62,  119 
Guglielmini,  128 
Gulistan,  102 
Gunther,  S.,  131 
Guyard,  S.,  82 

Habash,  9,  92 

Hager,  J.  (G.),  28,  32 

Ealliwell,  59,  85 

Hankel,  93 

1 1  ami  i  al-Kashid,  97,  106 

Bavet,  110 

Heath,  T.  L.,  125 


Hebrew  numerals,  127 

Hecatseus,  75 

Heiberg,  J.  L.,  55,  85,  148 

Heilbronner,  5 

Henry,    C,  5,   31,    55,   87,    120, 

135 
Heriger,  122 

Hermannus  Contractus,  123 
Herodotus,  76,  78 
Heyd,  75 
Higden,  136 
Hill,  G.  F.,  52,  139,  142 
Hillebrandt,  A.,  15,  74 
Hilprecht,  H.  V.,  28 
Hindu  forms,  early,  12 
Hindu  number  names,  42 
Hodder,  62 
Hoernle,  43,  49 
Holywood,  see  Sacrobosco 
Hopkins,  E.  W.,  12 
Horace,  79,  80 
Hosein    ibn    Mohammed    al-Ma- 

halli,  6 
Hostus,  M.,  56 
Howard,  H.  H.,  29 
Hrabanus  Maurus,  72 
Huart,  7 
Huet,  33 
Hugo,  H.,  57 
Humboldt,  A.  von,  62 
Huswirt,  58 

Iamblichus,  81 

Ibn  Abi  Ya'qub,  9 

Ibn  al-Adaml,  92 

Ibn  al-Banna,  93 

Ibn  Khordadbeh,  101,  106 

Ibn  Wahab,  103 

India,  history  of,  14 

writing  in,  18 
Indicopleustes,  83 
Indo-Bactrian  numerals,  19 


INDEX 


157 


Indrajl,  23 

Ishaq  ibn  Yusuf  al-Sardafl,  93 

Jacob  of  Florence,  57 

Jacquet,  E.,  38 

Jamshid,  56 

Jehan  Certain,  59 

Jetons,  58,  117 

Jevons,  F.  B.,  76 

Johannes  Hispalensis,  48,  88,  124 

John  of  Halifax,  see  Sacrobosco 

John  of  Luna,  see  Johannes  His- 
palensis 

Jordan,  L.,  58,  124 

Joseph  Ispanus  (Joseph  Sapiens), 
115 

Justinian,  104 

Kale,  M.  R.,  26 

Karabacek,  56 

Karpinski,  L.  C,  126,  134,  138 

Katyayana,  39 

Kaye,  C.  R.,  6,  16,  43,  46,  121 

Keane,  J.,  75,  82 

Keene,  H.  G.,  15 

Kern,  44 

KharosthI,  19,  20 

Khosru,  82,  91 

Kielhorn,  F.,  46,  47 

Kircher,  A.,  34 

Kitab  al-Fihrist,  see  Fihrist 

Kleinwachter,  32 

Klos,  62 

Kobel,  4,  58,  60,  119,  123 

Krumbacher,  K.,  57 

Kuekuck,  62,  133 

Kugler,  F.  X.,  51 

Lachmann,  85 
Lacouperie,  33,  35 
Lalitavistara,  15,  17 
Land,  G.,  57 


La  Roche,  61 

Lassen,  39 

Latyayana,  39 

Lebceuf,  135 

Leonardo  of  Pisa,  5,  10,  57,  64, 

74,  120,  128-133 
Lethaby,  W.  R.,  142 
Levi,  B.,  13 
Levias,  3 
Libri,  73,  85,  95 
Light  of  Asia,  16 
Luca  da  Firenze,  144 
Lucas,  128 

Mahabharata,  18 

Mahavlracarya,  53 

Malabar  numerals,  36 

Malayalam  numerals,  36 

Mannert,  81 

Margarita  Philosophica,  146 

Marie,  78 

Marquardt,  J.,  85 

Marshman,  J.  G,  17 

Martin,  T.  H.,  30,  62,  85,  113 

Martines,  D.  C,  58 

Mashallah,  3 

Maspero,  28 

Mauch,  142 

Maximus  Planudes,  2, 57, 66, 93, 120 

Megasthenes,  77 

Merchants,  114 

Meynard,  8 

Migne,  87 

Mikami,  Y.,  56 

Milanesi,  128 

Mohammed  ibn  'Abdallah,  92 

Mohammed  ibn  Ahmed,  6 

Mohammed  ibn  'Ali  'Abdi,  8 

Mohammed   ibn    Miisa,    see  Al- 

Khowarazmi 
Molinier,  123 
Monier-Williams,  17 


158 


THE  HINDU-ARABIC  NUMERALS 


Morley,  D.,  126 

Moroccan  numerals,  68,  119 
Mortet,  V.,  11 
Moseley,  C.  B.,  33 
Motahhar  ibn  Tahir,  7 
Mueller,  A.,  68 
Mumford,  J.  K.,  109 
Muwaffaq  al-DIn,  93 

Nabatean  forms,  21 

Nallino,  4,  54,  55 

Nagl,  A.,  55,  110,  113,  126 

Nana   Ghat   inscriptions,  20,  22, 

23,  40 
Narducci,  123 
Nasik  cave  inscriptions,  24 
Nazi!'  ibn  Yumn,  94 
Neander,  A.,  75 
Neophytos,  57,  62 
Neo-Pythagoreans,  64 
Nesselmann,  53 
Newman,  Cardinal,  96 
Newman,  F.  W.,  131 
Nbldeke,  Th.,  91 
Notation,  61 
Note,  61,  119 

Noviomagus,  45,  61,  119,  150 
Nidi,  61 
Numerals, 

Algerian,  68 

astrological,  150 

Brahmi,  19-22,  83 

early  ideas  of  origin,  1 

Hindu,  26 

Hindu,  classified,  19,  38 

Kharosthi,  19-22 

Moroccan,  68 

Nabatean,  21 

origin,  27,  30,  31,  37 

supposed  Arabic  origin,  2 

supposed  Babylonian  origin, 
28 


Numerals, 

supposed  Chaldean  and  Jew- 
ish origin,  3 
supposed  Chinese  origin,  28, 

32 
supposed  Egyptian  origin,  27, 

30,  69,  70 
supposed  Greek  origin,  33 
supposedPhoenician  origin,  32 
tables  of,  22-27,  36,   48,  50, 
69,  88,  140,  143,  145-148 

O'Creat,  5,  55,  119,  120 
Olleris,  110,  113 
Oppert,  G.,  14,  75 

Pali,  22 

Paficasiddhantika,  44 

Paravey,  32,  57 

Pataliputra,  77 

Patna,  77 

Patrick,  R.,  119 

Payne,  E.  J.,  106 

Pegolotti,  107 

Peletier,  2,  62 

Perrot,  80 

Persia,  66,  91,  107 

Pertz,  115 

Petrus  de  Dacia,  59,  61,  62 

Pez,  P.  B.,  117 

"  Philatelies,1''  75 

Phillips,  G,,  107 

Picavet,  105 

Pichler,  F.,  141 

Pihan,  A.  P.,  36 

Pisa,  128 

Place  value,  26,  42,  46,  48 

Planudes,  see  Maximus  Planudes 

Plimpton,  G.  A.,  56,  59,  85,  143, 

144,  145,  148 
Pliny,  76 
Polo,  N.  and  M.,  107 


INDEX 


159 


Prandel,  J.  G.,  54 

Prinsep,  J.,  20,  31 

Propertius,  80 

Prosdocimo  de'  Beldomandi,  137 

Prou,  143 

Ptolemy,  54,  78 

Putnam,  103 

Pythagoras,  63 

Pythagorean  numbers,  13 

Pytheas  of  Massilia,  76 

Rabbi  ben  Ezra,  60,  127 

Radulph  of  Laon,  60, 113, 118, 124 

Raets,  62 

Rainer,  see  Gemma  Frisius 

Ramayana,  18 

Ramus,  2,  41,  60,  61 

Raoul  Glaber,  123 

Rapson,  77 

Rauhfuss,  see  Dasypodius 

Raumer,  K.  von,  111 

Reclus,  E.,  14,  96,  130 

Recorde,  3,  58 

Reinaud,  67,  74,  80 

Reveillaud,  36 

Richer,  110,  112,  115 

Riese,  A.,  119 

Robertson,  81 

Robertus  Cestrensis,  97,  126 

Rodet,  5,  44 

Roediger,  J.,  68 

Rollandus,  144 

Romagnosi,  81 

Rosen,  F.,  5 

Rotula,  60 

Rudolff,  85 

Rudolph,  62,  67 

Ruffi,  150 

Sachau,  6 

Sacrobosco,  3,  58,  133 
Sacy,  S.  de,  66,  70 


Sa'di,  102 

Saka  inscriptions,  20 

Samu'il  ibn  Yahya,  93 

Sarada  characters,  55 

Savonne,  60 

Scaliger,  J.  C,  73 

Scheubel,  62 

Schlegel,  12 

Schmidt,  133 

Schonerus,  87,  110 

Schroeder,  L.  von,  13 

Scylax,  75 

Sedillot,  8,  34 

Senart,  20,  24,  25 

Sened  ibn  'All,  10,  98 

Sfortunati,  62,  150 

Shelley,  W.,  126 

Siamese  numerals,  36 

Siddhanta,  8,  18 

Sifr,  57 

Sigsboto,  55 

Sihab  al-Din,  67 

Silberberg,  60 

Simon,  13 

Sinan  ibn  al-Fath,  93 

Sindbad,  100 

Sindhind,  97 

Sipos,  60 

Sirr,  H.  C,  75 

Skeel,  C.  A.,  74 

Smith,  D.E.,  11, 17, 53, 86, 141, 143 

Smith,  V.  A.,  20,  35,  46,  47 

Smith,  Wm.,  75 

Smrti,  17 

Spain,  64,  65,  100 

Spitta-Bey,  5 

Sprenger,  94 

Srautasiitra,  39 

Steffens,  F.,  116 

Steinschneider,  5,  57,  65,  66,  98, 

126 
Stifel,  62 


160 


THE  HINDU-ARABIC  NUMERALS 


Subandhus,  44 

Suetonius,  80 

Suleiman,  100 

Sunya,  43,  53,  57 

Suter,  5,  9,  68,  69,  93,  116,  131 

Sutras,  13 

Sykes,  P.  M.,  75 

Sylvester  II,   see  Gerbert 

Symonds,  J.  A.,  129 

Tannery,  P.,  62,  84,  85 

Tartaglia,  4,  61 

Taylor,  I.,  19,  30 

Teca,  55,  61 

Tennent,  J.  E.,  75 

Texada,  60 

Theca,  58,  61 

Theophanes,  64 

Thibaut,  G.,  12,  13,  16,  44,  47 

Tibetan  numerals,  36 

Timotheus,  103 

Tonstall,  C.,  3,  61 

Trenchant,  60 

Treutlein,  5,  63,  123 

Trevisa,  136 

Treviso  arithmetic,  145 

Trivium  and  quadrivium,  73 

Tsin,  56 

Tunis,  65 

Turchill,  88,  118,  123 

Tumour,  G.,  75 

Tziphra,  57,  62 

T^l<pPa,  55,  57,  62 

Tzwivel,  61,  118,  145 

Ujjain,  32 
Unger,  133 
Upanishads,  12 
Usk,  121 

Valla,  G.,  61 

Van  dor  Schuere,  62 


Varaha-Mihira,  39,  44,  78 

Vasavadatta,  44 

Vaux,  Carra  de,  9,  74 

Vaux,  W.  S.  W.,  91 

Vedahgas,  17 

Vedas,  12,  15,  17 

Vergil,  80 

Vincent,  A.  J.  H.,  57 

Vogt,  13 

Voizot,  P.,  36 

Vossius,  4,  76,  81,  84 

Wallis,  3,  62,  84,  116 
Wappler,  E.,  54,  126 
Waschke,  H.,  2,  93 
Wattenbach,  143 
Weber,  A.,  31 
Weidler,  I.  F.,  34,  66 
Weidler,  I.  F.  and  G.  I.,  63,  66 
Weissenborn,  85,  110 
Wertheim,  G.,  57,  61 
Whitney,  W.  D..  13 
Wilford,  F.,  75 
Wilkens,  62 
Wilkinson,  J.  G.,  70 
Willichius,  3 
Woepcke,  3,  6,  42,  63,  64,  65,  67, 

69,  70,  94,  113,  138 
Wolack,  G.,  54 
Woodruff,  C.  E.,  32 
Word   and   letter   numerals,  38, 

44 
Wustenfeld,  74 

Yule,  H.,  107 

Zephirum,  57,  58 

Zephyr,  59 

Zepiro,  58 

Zero,  26,  38,  40,  43,  45,  50,  51-62, 

67 
Zeuero,  58 


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7iI^i«MAYM^ECAL7J^AFTER  7  DAYS 
2  month  loans  may  be  renewed  by  calling 

,-ieXs  may  be  recharged  by  bringing  books 

Renewed  recharges  may  be  made  4  days 
prior  to  due  date 


DUE  AS  STAMPED  BELOW 


— MAR- — ZUUD" 


X 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


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